गुणोत्तर श्रेणी \(5,15,45,135,\ldots\) का तेरहवाँ पद क्या है?
What is the thirteenth term of the geometric progression \(5,15,45,135,\ldots\)?
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A (1328605)
B (1771470)
C (2657205)
D (5314410)
Explanation opens after your attempt
Correct Answer
C. (2657205)
Step 1
Concept
Here (a=5) and (r=3), so \(a_{13}=5\cdot3^{12}=2657205\). In exams, use \(a_n=ar^{n-1}\).
Step 2
Why this answer is correct
The correct answer is C. (2657205). Here (a=5) and (r=3), so \(a_{13}=5\cdot3^{12}=2657205\). In exams, use \(a_n=ar^{n-1}\).
Step 3
Exam Tip
यहाँ (a=5) और (r=3) है, इसलिए \(a_{13}=5\cdot3^{12}=2657205\) है। परीक्षा में \(a_n=ar^{n-1}\) लगाएँ।
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यदि किसी गुणोत्तर श्रेणी में (a=7) और (r=4) है, तो \(a_7\) का मान क्या होगा?
If a geometric progression has (a=7) and (r=4), what is the value of \(a_7\)?
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A (14336)
B (28672)
C (32768)
D (57344)
Explanation opens after your attempt
Correct Answer
B. (28672)
Step 1
Concept
\(a_7=7\cdot4^6=28672\). In exams, use \(r^6\) for the seventh term.
Step 2
Why this answer is correct
The correct answer is B. (28672). \(a_7=7\cdot4^6=28672\). In exams, use \(r^6\) for the seventh term.
Step 3
Exam Tip
\(a_7=7\cdot4^6=28672\) है। परीक्षा में सातवें पद के लिए \(r^6\) लगाएँ।
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गुणोत्तर श्रेणी \(729,243,81,27,\ldots\) का आठवाँ पद क्या है?
What is the eighth term of the geometric progression \(729,243,81,27,\ldots\)?
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A (1)
B \(\frac{1}{3}\)
C \(\frac{1}{9}\)
D (3)
Explanation opens after your attempt
Correct Answer
B. \(\frac{1}{3}\)
Step 1
Concept
Here \(r=\frac{1}{3}\), so (a_8=729\cdot\left\(\frac{1}{3}\right\)7 =\frac{1}{3}). In exams, apply fractional ratios carefully in decreasing GPs.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{1}{3}\). Here \(r=\frac{1}{3}\), so (a_8=729\cdot\left\(\frac{1}{3}\right\)7 =\frac{1}{3}). In exams, apply fractional ratios carefully in decreasing GPs.
Step 3
Exam Tip
यहाँ \(r=\frac{1}{3}\) है, इसलिए (a_8=729\cdot\left\(\frac{1}{3}\right\)7 =\frac{1}{3}) है। परीक्षा में घटती GP में भिन्न अनुपात सावधानी से लगाएँ।
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यदि \(a_n=8\cdot2^{n-1}\) है, तो (4096) कौन-सा पद होगा?
If \(a_n=8\cdot2^{n-1}\), which term will be (4096)?
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A आठवाँ पद / (8)th term
B नौवाँ पद / (9)th term
C दसवाँ पद / (10)th term
D ग्यारहवाँ पद / (11)th term
Explanation opens after your attempt
Correct Answer
C. दसवाँ पद / (10)th term
Step 1
Concept
From \(8\cdot2^{n-1}=4096\), \(2^{n-1}=512=2^9\), so (n=10). In exams, equate powers to find the term number.
Step 2
Why this answer is correct
The correct answer is C. दसवाँ पद / (10)th term. From \(8\cdot2^{n-1}=4096\), \(2^{n-1}=512=2^9\), so (n=10). In exams, equate powers to find the term number.
Step 3
Exam Tip
\(8\cdot2^{n-1}=4096\) से \(2^{n-1}=512=2^9\), इसलिए (n=10) है। परीक्षा में घातों को बराबर करके पद संख्या निकालें।
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किसी गुणोत्तर श्रेणी में पहला पद (9) और पाँचवाँ पद (2304) है। यदि (r) धनात्मक है, तो (r) क्या है?
In a geometric progression, the first term is (9) and the fifth term is (2304). If (r) is positive, what is (r)?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
From \(2304=9r^4\), \(r^4=256=4^4\), so (r=4). In exams, remember \(r^4\) for the fifth term.
Step 2
Why this answer is correct
The correct answer is C. (4). From \(2304=9r^4\), \(r^4=256=4^4\), so (r=4). In exams, remember \(r^4\) for the fifth term.
Step 3
Exam Tip
\(2304=9r^4\) से \(r^4=256=4^4\), इसलिए (r=4) है। परीक्षा में पाँचवें पद के लिए \(r^4\) याद रखें।
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गुणोत्तर श्रेणी \(14,42,126,378,\ldots\) का सामान्य पद क्या है?
What is the general term of the geometric progression \(14,42,126,378,\ldots\)?
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A \(a_n=14\cdot3^{n-1}\)
B \(a_n=3\cdot14^{n-1}\)
C \(a_n=14n+28\)
D \(a_n=42n\)
Explanation opens after your attempt
Correct Answer
A. \(a_n=14\cdot3^{n-1}\)
Step 1
Concept
The first term is (14) and the ratio is (3), so \(a_n=14\cdot3^{n-1}\). In exams, keep (a) and (r) correct in \(ar^{n-1}\).
Step 2
Why this answer is correct
The correct answer is A. \(a_n=14\cdot3^{n-1}\). The first term is (14) and the ratio is (3), so \(a_n=14\cdot3^{n-1}\). In exams, keep (a) and (r) correct in \(ar^{n-1}\).
Step 3
Exam Tip
पहला पद (14) और अनुपात (3) है, इसलिए \(a_n=14\cdot3^{n-1}\) है। परीक्षा में \(ar^{n-1}\) में (a) और (r) सही रखें।
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गुणोत्तर श्रेणी \(4,12,36,108,\ldots\) के पहले (6) पदों का योग क्या है?
What is the sum of the first (6) terms of the geometric progression \(4,12,36,108,\ldots\)?
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A (1456)
B (1458)
C (1460)
D (1464)
Explanation opens after your attempt
Step 1
Concept
(S_6=\frac{4\(3^6-1\)}{3-1}=1456). In exams, use (\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 2
Why this answer is correct
The correct answer is A. (1456). (S_6=\frac{4\(3^6-1\)}{3-1}=1456). In exams, use (\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 3
Exam Tip
(S_6=\frac{4\(3^6-1\)}{3-1}=1456) है। परीक्षा में (r>1) के लिए (\frac{a\(r^n-1\)}{r-1}) लगाएँ।
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यदि (18,x,200) गुणोत्तर श्रेणी में हैं और (x>0) है, तो (x) क्या होगा?
If (18,x,200) are in geometric progression and (x>0), what is (x)?
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A (40)
B (50)
C (60)
D (70)
Explanation opens after your attempt
Step 1
Concept
For the middle term, \(x^2=18\cdot200=3600\), so (x=60). In exams, the square of the middle GP term equals the product of the outer terms.
Step 2
Why this answer is correct
The correct answer is C. (60). For the middle term, \(x^2=18\cdot200=3600\), so (x=60). In exams, the square of the middle GP term equals the product of the outer terms.
Step 3
Exam Tip
मध्य पद के लिए \(x^2=18\cdot200=3600\), इसलिए (x=60) है। परीक्षा में तीन GP पदों में मध्य पद का वर्ग बाहरी पदों के गुणनफल के बराबर होता है।
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गुणोत्तर श्रेणी \(6,30,150,750,\ldots\) के पहले (5) पदों का योग क्या है?
What is the sum of the first (5) terms of the geometric progression \(6,30,150,750,\ldots\)?
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A (4680)
B (4686)
C (4690)
D (4695)
Explanation opens after your attempt
Step 1
Concept
The first five terms are (6,30,150,750,3750), and their sum is (4686). In exams, add terms carefully when the ratio is large.
Step 2
Why this answer is correct
The correct answer is B. (4686). The first five terms are (6,30,150,750,3750), and their sum is (4686). In exams, add terms carefully when the ratio is large.
Step 3
Exam Tip
पहले पाँच पद (6,30,150,750,3750) हैं और योग (4686) है। परीक्षा में बड़े अनुपात में पदों को सावधानी से जोड़ें।
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यदि \(a_6=1024\) और (r=2) है, तो पहला पद (a) क्या होगा?
If \(a_6=1024\) and (r=2), what is the first term (a)?
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A (16)
B (24)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
\(a_6=ar^5\), so \(1024=a\cdot32\) and (a=32). In exams, divide the sixth term by \(r^5\).
Step 2
Why this answer is correct
The correct answer is C. (32). \(a_6=ar^5\), so \(1024=a\cdot32\) and (a=32). In exams, divide the sixth term by \(r^5\).
Step 3
Exam Tip
\(a_6=ar^5\), इसलिए \(1024=a\cdot32\) और (a=32) है। परीक्षा में छठे पद से \(r^5\) भाग दें।
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गुणोत्तर श्रेणी \(216,72,24,8,\ldots\) का सामान्य पद क्या है?
What is the general term of the geometric progression \(216,72,24,8,\ldots\)?
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A \(a_n=216\cdot3^{n-1}\)
B (a_n=216\cdot\left\(\frac{1}{3}\right\)^{n-1})
C \(a_n=216-72n\)
D \(a_n=72n\)
Explanation opens after your attempt
Correct Answer
B. (a_n=216\cdot\left\(\frac{1}{3}\right\)^{n-1})
Step 1
Concept
The first term is (216) and the ratio is \(\frac{1}{3}\), so the correct rule is (216\cdot\left\(\frac{1}{3}\right\)^{n-1}). In exams, write the fractional ratio in a decreasing GP.
Step 2
Why this answer is correct
The correct answer is B. (a_n=216\cdot\left\(\frac{1}{3}\right\)^{n-1}). The first term is (216) and the ratio is \(\frac{1}{3}\), so the correct rule is (216\cdot\left\(\frac{1}{3}\right\)^{n-1}). In exams, write the fractional ratio in a decreasing GP.
Step 3
Exam Tip
पहला पद (216) और अनुपात \(\frac{1}{3}\) है, इसलिए सही नियम (216\cdot\left\(\frac{1}{3}\right\)^{n-1}) है। परीक्षा में घटती GP में भिन्न अनुपात लिखें।
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किस गुणोत्तर श्रेणी में (a=2) और \(a_7=1458\) है। यदि (r) धनात्मक है, तो (r) क्या है?
For a geometric progression with (a=2) and \(a_7=1458\), if (r) is positive, what is (r)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
From \(1458=2r^6\), \(r^6=729=3^6\), so (r=3). In exams, \(r^6\) is used for the seventh term.
Step 2
Why this answer is correct
The correct answer is B. (3). From \(1458=2r^6\), \(r^6=729=3^6\), so (r=3). In exams, \(r^6\) is used for the seventh term.
Step 3
Exam Tip
\(1458=2r^6\) से \(r^6=729=3^6\), इसलिए (r=3) है। परीक्षा में सातवें पद के लिए \(r^6\) लगता है।
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गुणोत्तर श्रेणी \(8,32,128,512,\ldots\) में (8192) कौन-सा पद है?
In the geometric progression \(8,32,128,512,\ldots\), which term is (8192)?
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A पाँचवाँ पद / (5)th term
B छठा पद / (6)th term
C सातवाँ पद / (7)th term
D आठवाँ पद / (8)th term
Explanation opens after your attempt
Correct Answer
B. छठा पद / (6)th term
Step 1
Concept
From \(8\cdot4^{n-1}=8192\), \(4^{n-1}=1024=4^5\), so (n=6). In exams, equate powers to find (n).
Step 2
Why this answer is correct
The correct answer is B. छठा पद / (6)th term. From \(8\cdot4^{n-1}=8192\), \(4^{n-1}=1024=4^5\), so (n=6). In exams, equate powers to find (n).
Step 3
Exam Tip
\(8\cdot4^{n-1}=8192\) से \(4^{n-1}=1024=4^5\), इसलिए (n=6) है। परीक्षा में घातों को बराबर करके (n) निकालें।
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यदि (a=256) और \(r=\frac{1}{2}\) है, तो \(a_9\) का मान क्या होगा?
If (a=256) and \(r=\frac{1}{2}\), what is the value of \(a_9\)?
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A (1)
B (2)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
(a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1). In exams, apply powers of fractional ratios carefully.
Step 2
Why this answer is correct
The correct answer is A. (1). (a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1). In exams, apply powers of fractional ratios carefully.
Step 3
Exam Tip
(a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1) है। परीक्षा में भिन्न अनुपात के घात सावधानी से लगाएँ।
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यदि गुणोत्तर श्रेणी के पहले (4) पदों का योग (780) है और (a=12) तथा (r=4) है, तो कथन कैसा है?
If the sum of the first (4) terms of a geometric progression is (780) with (a=12) and (r=4), what is the statement?
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A सही क्योंकि \(S_4=780\) / True because \(S_4=780\)
B गलत क्योंकि \(S_4=760\) / False because \(S_4=760\)
C गलत क्योंकि \(S_4=768\) / False because \(S_4=768\)
D गलत क्योंकि \(S_4=800\) / False because \(S_4=800\)
Explanation opens after your attempt
Correct Answer
A. सही क्योंकि \(S_4=780\) / True because \(S_4=780\)
Step 1
Concept
(S_4=\frac{12\(4^4-1\)}{4-1}=1020), so the statement is false. In exams, verify statement-type questions carefully.
Step 2
Why this answer is correct
The correct answer is A. सही क्योंकि \(S_4=780\) / True because \(S_4=780\). (S_4=\frac{12\(4^4-1\)}{4-1}=1020), so the statement is false. In exams, verify statement-type questions carefully.
Step 3
Exam Tip
(S_4=\frac{12\(4^4-1\)}{4-1}=1020) नहीं; सीधे योग (12+48+192+768=1020) है, इसलिए कथन गलत है।
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गुणोत्तर श्रेणी \(5,20,80,320,\ldots\) के पहले (6) पदों का योग क्या है?
What is the sum of the first (6) terms of the geometric progression \(5,20,80,320,\ldots\)?
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A (6825)
B (6820)
C (6815)
D (6800)
Explanation opens after your attempt
Step 1
Concept
(S_6=\frac{5\(4^6-1\)}{4-1}=6825). In exams, use (\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 2
Why this answer is correct
The correct answer is A. (6825). (S_6=\frac{5\(4^6-1\)}{4-1}=6825). In exams, use (\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 3
Exam Tip
(S_6=\frac{5\(4^6-1\)}{4-1}=6825) है। परीक्षा में (r>1) के लिए (\frac{a\(r^n-1\)}{r-1}) उपयोग करें।
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गुणोत्तर श्रेणी में \(a_3=36\) और \(a_7=576\) है। यदि (r) धनात्मक है, तो (r) क्या है?
In a geometric progression \(a_3=36\) and \(a_7=576\). If (r) is positive, what is (r)?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(\frac{a_7}{a_3}=r^4=\frac{576}{36}=16\), so (r=2). In exams, the ratio of distant terms gives a power based on the position gap.
Step 2
Why this answer is correct
The correct answer is A. (2). \(\frac{a_7}{a_3}=r^4=\frac{576}{36}=16\), so (r=2). In exams, the ratio of distant terms gives a power based on the position gap.
Step 3
Exam Tip
\(\frac{a_7}{a_3}=r^4=\frac{576}{36}=16\), इसलिए (r=2) है। परीक्षा में दूर के पदों का अनुपात पद-अंतर की घात देता है।
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यदि \(a_7=2187\) और (r=3) है, तो \(a_1\) का मान क्या है?
If \(a_7=2187\) and (r=3), what is the value of \(a_1\)?
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A (1)
B (2)
C (3)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(a_7=a\cdot3^6\), so \(a=2187\div729=3\). In exams, divide the seventh term by \(r^6\).
Step 2
Why this answer is correct
The correct answer is C. (3). \(a_7=a\cdot3^6\), so \(a=2187\div729=3\). In exams, divide the seventh term by \(r^6\).
Step 3
Exam Tip
\(a_7=a\cdot3^6\), इसलिए \(a=2187\div729=3\) है। परीक्षा में सातवें पद से \(r^6\) भाग दें।
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गुणोत्तर श्रेणी \(9,27,81,243,\ldots\) का \(a_9\div a_5\) क्या होगा?
What is \(a_9\div a_5\) for the geometric progression \(9,27,81,243,\ldots\)?
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A (27)
B (81)
C (243)
D (729)
Explanation opens after your attempt
Step 1
Concept
The position gap is (4) and (r=3), so the ratio is \(3^4=81\). In exams, use \(\frac{a_m}{a_n}=r^{m-n}\).
Step 2
Why this answer is correct
The correct answer is B. (81). The position gap is (4) and (r=3), so the ratio is \(3^4=81\). In exams, use \(\frac{a_m}{a_n}=r^{m-n}\).
Step 3
Exam Tip
पद-संख्या का अंतर (4) है और (r=3), इसलिए अनुपात \(3^4=81\) है। परीक्षा में \(\frac{a_m}{a_n}=r^{m-n}\) लगाएँ।
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यदि \(a_n=6\cdot4^{n-1}\) है, तो पहले (5) पदों का योग क्या होगा?
If \(a_n=6\cdot4^{n-1}\), what is the sum of the first (5) terms?
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A (2046)
B (2050)
C (2052)
D (2056)
Explanation opens after your attempt
Step 1
Concept
Here (a=6) and (r=4), so (S_5=\frac{6\(4^5-1\)}{4-1}=2046). In exams, identify (a) and (r) from the general term.
Step 2
Why this answer is correct
The correct answer is A. (2046). Here (a=6) and (r=4), so (S_5=\frac{6\(4^5-1\)}{4-1}=2046). In exams, identify (a) and (r) from the general term.
Step 3
Exam Tip
यहाँ (a=6) और (r=4) है, इसलिए (S_5=\frac{6\(4^5-1\)}{4-1}=2046) है। परीक्षा में सामान्य पद से (a) और (r) पहचानें।
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गुणोत्तर श्रेणी \(7,21,63,\ldots\) का वह पद कौन-सा है जो (5103) है?
Which term of the geometric progression \(7,21,63,\ldots\) is (5103)?
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A पाँचवाँ पद / (5)th term
B छठा पद / (6)th term
C सातवाँ पद / (7)th term
D आठवाँ पद / (8)th term
Explanation opens after your attempt
Correct Answer
C. सातवाँ पद / (7)th term
Step 1
Concept
From \(7\cdot3^{n-1}=5103\), \(3^{n-1}=729=3^6\), so (n=7). In exams, compare powers to find the term number.
Step 2
Why this answer is correct
The correct answer is C. सातवाँ पद / (7)th term. From \(7\cdot3^{n-1}=5103\), \(3^{n-1}=729=3^6\), so (n=7). In exams, compare powers to find the term number.
Step 3
Exam Tip
\(7\cdot3^{n-1}=5103\) से \(3^{n-1}=729=3^6\), इसलिए (n=7) है। परीक्षा में पद संख्या के लिए घात की तुलना करें।
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यदि \(a_2=20\) और \(a_6=5120\) है, तो (r) का धनात्मक मान क्या है?
If \(a_2=20\) and \(a_6=5120\), what is the positive value of (r)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
\(\frac{a_6}{a_2}=r^4=\frac{5120}{20}=256\), so (r=4). In exams, a gap of (4) positions gives \(r^4\).
Step 2
Why this answer is correct
The correct answer is C. (4). \(\frac{a_6}{a_2}=r^4=\frac{5120}{20}=256\), so (r=4). In exams, a gap of (4) positions gives \(r^4\).
Step 3
Exam Tip
\(\frac{a_6}{a_2}=r^4=\frac{5120}{20}=256\), इसलिए (r=4) है। परीक्षा में पद-अंतर (4) हो तो \(r^4\) बनेगा।
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गुणोत्तर श्रेणी \(640,320,160,80,\ldots\) में (5) कौन-सा पद है?
In the geometric progression \(640,320,160,80,\ldots\), which term is (5)?
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A सातवाँ पद / (7)th term
B आठवाँ पद / (8)th term
C नौवाँ पद / (9)th term
D दसवाँ पद / (10)th term
Explanation opens after your attempt
Correct Answer
B. आठवाँ पद / (8)th term
Step 1
Concept
Each term is multiplied by \(\frac{1}{2}\), and the terms are (640,320,160,80,40,20,10,5). In exams, you can also check a decreasing GP in order.
Step 2
Why this answer is correct
The correct answer is B. आठवाँ पद / (8)th term. Each term is multiplied by \(\frac{1}{2}\), and the terms are (640,320,160,80,40,20,10,5). In exams, you can also check a decreasing GP in order.
Step 3
Exam Tip
हर बार \(\frac{1}{2}\) से गुणा होता है और पद (640,320,160,80,40,20,10,5) हैं। परीक्षा में घटती GP को क्रम से भी जाँच सकते हैं।
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यदि (a=13) और \(a_6=416\) है तथा (r) धनात्मक है, तो (r) क्या होगा?
If (a=13) and \(a_6=416\), and (r) is positive, what is (r)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
From \(416=13r^5\), \(r^5=32=2^5\), so (r=2). In exams, use \(r^5\) for the sixth term.
Step 2
Why this answer is correct
The correct answer is A. (2). From \(416=13r^5\), \(r^5=32=2^5\), so (r=2). In exams, use \(r^5\) for the sixth term.
Step 3
Exam Tip
\(416=13r^5\) से \(r^5=32=2^5\), इसलिए (r=2) है। परीक्षा में छठे पद के लिए \(r^5\) लगाएँ।
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गुणोत्तर श्रेणी \(11,33,99,\ldots\) के पहले (6) पदों का योग क्या है?
What is the sum of the first (6) terms of the geometric progression \(11,33,99,\ldots\)?
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A (4004)
B (4000)
C (3993)
D (4015)
Explanation opens after your attempt
Step 1
Concept
(S_6=\frac{11\(3^6-1\)}{3-1}=4004). In exams, first calculate the value of \(3^6\).
Step 2
Why this answer is correct
The correct answer is A. (4004). (S_6=\frac{11\(3^6-1\)}{3-1}=4004). In exams, first calculate the value of \(3^6\).
Step 3
Exam Tip
(S_6=\frac{11\(3^6-1\)}{3-1}=4004) है। परीक्षा में पहले \(3^6\) का मान निकालें।
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किस गुणोत्तर श्रेणी में \(a_3=80\) और \(a_6=5120\) है। यदि (r) धनात्मक है, तो \(a_1\) क्या है?
In a geometric progression \(a_3=80\) and \(a_6=5120\). If (r) is positive, what is \(a_1\)?
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A (2)
B (5)
C (10)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(\frac{5120}{80}=64=r^3\), so (r=4) and \(a_1=\frac{80}{4^2}=5\). In exams, find (r) first and then the first term.
Step 2
Why this answer is correct
The correct answer is B. (5). \(\frac{5120}{80}=64=r^3\), so (r=4) and \(a_1=\frac{80}{4^2}=5\). In exams, find (r) first and then the first term.
Step 3
Exam Tip
\(\frac{5120}{80}=64=r^3\) से (r=4) और \(a_1=\frac{80}{4^2}=5\) है। परीक्षा में पहले (r) निकालें फिर पहला पद।
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यदि गुणोत्तर श्रेणी \(x,,4x,,16x,\ldots\) में पाँचवाँ पद (1280) है, तो (x) क्या है?
If the fifth term of the geometric progression \(x,,4x,,16x,\ldots\) is (1280), what is (x)?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The fifth term is \(x\cdot4^4=256x=1280\), so (x=5). In exams, put the algebraic first term in \(ar^{n-1}\) too.
Step 2
Why this answer is correct
The correct answer is C. (5). The fifth term is \(x\cdot4^4=256x=1280\), so (x=5). In exams, put the algebraic first term in \(ar^{n-1}\) too.
Step 3
Exam Tip
पाँचवाँ पद \(x\cdot4^4=256x=1280\) है, इसलिए (x=5) है। परीक्षा में बीजीय पहले पद को भी \(ar^{n-1}\) में रखें।
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गुणोत्तर श्रेणी \(3,12,48,\ldots\) में (12288) तक कितने पद हैं?
How many terms are there up to (12288) in the geometric progression \(3,12,48,\ldots\)?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(3\cdot4^{n-1}=12288\), so \(4^{n-1}=4096=4^6\) and (n=7). In exams, equate the last term to the general term.
Step 2
Why this answer is correct
The correct answer is B. (7). \(3\cdot4^{n-1}=12288\), so \(4^{n-1}=4096=4^6\) and (n=7). In exams, equate the last term to the general term.
Step 3
Exam Tip
\(3\cdot4^{n-1}=12288\) से \(4^{n-1}=4096=4^6\), इसलिए (n=7) है। परीक्षा में अंतिम पद को सामान्य पद के बराबर रखें।
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यदि किसी गुणोत्तर श्रेणी के पहले (4) पद (5,25,125,625) हैं, तो \(S_4\) क्या है?
If the first (4) terms of a geometric progression are (5,25,125,625), what is \(S_4\)?
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A (760)
B (775)
C (780)
D (800)
Explanation opens after your attempt
Step 1
Concept
The sum is (5+25+125+625=780). In exams, direct addition is also fast for small (n).
Step 2
Why this answer is correct
The correct answer is C. (780). The sum is (5+25+125+625=780). In exams, direct addition is also fast for small (n).
Step 3
Exam Tip
योग (5+25+125+625=780) है। परीक्षा में छोटे (n) के लिए सीधे जोड़ना भी तेज होता है।
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गुणोत्तर श्रेणी \(9,36,144,\ldots\) का कौन-सा पद (9216) है?
Which term of the geometric progression \(9,36,144,\ldots\) is (9216)?
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A पाँचवाँ पद / (5)th term
B छठा पद / (6)th term
C सातवाँ पद / (7)th term
D आठवाँ पद / (8)th term
Explanation opens after your attempt
Correct Answer
B. छठा पद / (6)th term
Step 1
Concept
From \(9\cdot4^{n-1}=9216\), \(4^{n-1}=1024=4^5\), so (n=6). In exams, equate powers to find the term number.
Step 2
Why this answer is correct
The correct answer is B. छठा पद / (6)th term. From \(9\cdot4^{n-1}=9216\), \(4^{n-1}=1024=4^5\), so (n=6). In exams, equate powers to find the term number.
Step 3
Exam Tip
\(9\cdot4^{n-1}=9216\) से \(4^{n-1}=1024=4^5\), इसलिए (n=6) है। परीक्षा में घात बराबर करके पद संख्या निकालें।
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यदि \(a_1=6\) और (r=-2) है, तो \(a_7\) क्या होगा?
If \(a_1=6\) and (r=-2), what is \(a_7\)?
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A (384)
B (-384)
C (192)
D (-192)
Explanation opens after your attempt
Step 1
Concept
(a_7=6(-2)6 =384). In exams, an even power of a negative ratio is positive.
Step 2
Why this answer is correct
The correct answer is A. (384). (a_7=6(-2)6 =384). In exams, an even power of a negative ratio is positive.
Step 3
Exam Tip
(a_7=6(-2)6 =384) है। परीक्षा में ऋणात्मक अनुपात की सम घात धनात्मक होती है।
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गुणोत्तर श्रेणी \(3,18,108,648,\ldots\) के पहले (4) पदों का योग क्या है?
What is the sum of the first (4) terms of the geometric progression \(3,18,108,648,\ldots\)?
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A (777)
B (780)
C (783)
D (786)
Explanation opens after your attempt
Step 1
Concept
The sum of the first four terms is (3+18+108+648=777). In exams, direct addition is safe when the number of terms is small.
Step 2
Why this answer is correct
The correct answer is A. (777). The sum of the first four terms is (3+18+108+648=777). In exams, direct addition is safe when the number of terms is small.
Step 3
Exam Tip
पहले चार पदों का योग (3+18+108+648=777) है। परीक्षा में पद कम हों तो सीधे जोड़ना सुरक्षित है।
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यदि \(a_4=96\) और (r=2) है, तो \(a_{10}\) का मान क्या होगा?
If \(a_4=96\) and (r=2), what is the value of \(a_{10}\)?
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A (3072)
B (6144)
C (12288)
D (24576)
Explanation opens after your attempt
Step 1
Concept
There are (6) steps from the fourth to the tenth term, so \(a_{10}=96\cdot2^6=6144\). In exams, count the position gap.
Step 2
Why this answer is correct
The correct answer is B. (6144). There are (6) steps from the fourth to the tenth term, so \(a_{10}=96\cdot2^6=6144\). In exams, count the position gap.
Step 3
Exam Tip
चौथे से दसवें पद तक (6) कदम हैं, इसलिए \(a_{10}=96\cdot2^6=6144\) है। परीक्षा में पद-संख्या का अंतर गिनें।
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गुणोत्तर श्रेणी \(810,270,90,\ldots\) में (10) कौन-सा पद है?
In the geometric progression \(810,270,90,\ldots\), which term is (10)?
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A चौथा पद / (4)th term
B पाँचवाँ पद / (5)th term
C छठा पद / (6)th term
D सातवाँ पद / (7)th term
Explanation opens after your attempt
Correct Answer
B. पाँचवाँ पद / (5)th term
Step 1
Concept
Each term is multiplied by \(\frac{1}{3}\), so the terms are (810,270,90,30,10). In exams, you can also check a decreasing GP in order.
Step 2
Why this answer is correct
The correct answer is B. पाँचवाँ पद / (5)th term. Each term is multiplied by \(\frac{1}{3}\), so the terms are (810,270,90,30,10). In exams, you can also check a decreasing GP in order.
Step 3
Exam Tip
हर बार \(\frac{1}{3}\) से गुणा होता है, इसलिए पद (810,270,90,30,10) हैं। परीक्षा में घटती GP को क्रम से भी जाँच सकते हैं।
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यदि (a=10), (r=3) और \(S_5=1210\) है, तो कथन कैसा है?
If (a=10), (r=3), and \(S_5=1210\), what is the statement?
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A सही क्योंकि \(S_5=1210\) / True because \(S_5=1210\)
B गलत क्योंकि \(S_5=1200\) / False because \(S_5=1200\)
C गलत क्योंकि \(S_5=1220\) / False because \(S_5=1220\)
D गलत क्योंकि \(S_5=2430\) / False because \(S_5=2430\)
Explanation opens after your attempt
Correct Answer
A. सही क्योंकि \(S_5=1210\) / True because \(S_5=1210\)
Step 1
Concept
(S_5=\frac{10\(3^5-1\)}{3-1}=1210). In exams, confirm statement-type questions with the formula.
Step 2
Why this answer is correct
The correct answer is A. सही क्योंकि \(S_5=1210\) / True because \(S_5=1210\). (S_5=\frac{10\(3^5-1\)}{3-1}=1210). In exams, confirm statement-type questions with the formula.
Step 3
Exam Tip
(S_5=\frac{10\(3^5-1\)}{3-1}=1210) है। परीक्षा में कथन-प्रकार प्रश्न में सूत्र से पुष्टि करें।
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गुणोत्तर श्रेणी \(4,16,64,\ldots\) के \(a_5+a_6\) का मान क्या है?
What is the value of \(a_5+a_6\) for the geometric progression \(4,16,64,\ldots\)?
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A (4096)
B (5120)
C (6144)
D (8192)
Explanation opens after your attempt
Step 1
Concept
\(a_5=1024\) and \(a_6=4096\), so the sum is (5120). In exams, find both terms separately and add.
Step 2
Why this answer is correct
The correct answer is B. (5120). \(a_5=1024\) and \(a_6=4096\), so the sum is (5120). In exams, find both terms separately and add.
Step 3
Exam Tip
\(a_5=1024\) और \(a_6=4096\), इसलिए योग (5120) है। परीक्षा में दोनों पद अलग-अलग निकालकर जोड़ें।
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यदि \(a_n=162\left(\frac{1}{3}\right)^{n-1}\) है, तो कौन-सा पद (2) के बराबर होगा?
If \(a_n=162\left(\frac{1}{3}\right)^{n-1}\), which term will be equal to (2)?
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A चौथा पद / (4)th term
B पाँचवाँ पद / (5)th term
C छठा पद / (6)th term
D सातवाँ पद / (7)th term
Explanation opens after your attempt
Correct Answer
B. पाँचवाँ पद / (5)th term
Step 1
Concept
From \(162\left(\frac{1}{3}\right)^{n-1}=2\), \(\left(\frac{1}{3}\right)^{n-1}=\frac{1}{81}\), so (n=5). In exams, simplify fractions first.
Step 2
Why this answer is correct
The correct answer is B. पाँचवाँ पद / (5)th term. From \(162\left(\frac{1}{3}\right)^{n-1}=2\), \(\left(\frac{1}{3}\right)^{n-1}=\frac{1}{81}\), so (n=5). In exams, simplify fractions first.
Step 3
Exam Tip
\(162\left(\frac{1}{3}\right)^{n-1}=2\) से \(\left(\frac{1}{3}\right)^{n-1}=\frac{1}{81}\), इसलिए (n=5) है। परीक्षा में भिन्नों को पहले सरल करें।
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गुणोत्तर श्रेणी \(7,14,28,\ldots\) के पहले (9) पदों का योग क्या है?
What is the sum of the first (9) terms of the geometric progression \(7,14,28,\ldots\)?
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A (3577)
B (3584)
C (3591)
D (3600)
Explanation opens after your attempt
Step 1
Concept
(S_9=7\(2^9-1\)=3577). In exams, remembering \(2^9=512\) is useful.
Step 2
Why this answer is correct
The correct answer is A. (3577). (S_9=7\(2^9-1\)=3577). In exams, remembering \(2^9=512\) is useful.
Step 3
Exam Tip
(S_9=7\(2^9-1\)=3577) है। परीक्षा में \(2^9=512\) याद रखना उपयोगी है।
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यदि किसी गुणोत्तर श्रेणी में \(a_5=96\) और \(a_9=1536\) है, तो (r) का धनात्मक मान क्या है?
If a geometric progression has \(a_5=96\) and \(a_9=1536\), what is the positive value of (r)?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(\frac{1536}{96}=16=r^4\), so (r=2). In exams, a position gap of (4) gives \(r^4\).
Step 2
Why this answer is correct
The correct answer is A. (2). \(\frac{1536}{96}=16=r^4\), so (r=2). In exams, a position gap of (4) gives \(r^4\).
Step 3
Exam Tip
\(\frac{1536}{96}=16=r^4\), इसलिए (r=2) है। परीक्षा में पद-संख्या का अंतर (4) होने पर \(r^4\) मिलेगा।
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गुणोत्तर श्रेणी \(15,45,135,\ldots\) में \(a_8\) क्या है?
What is \(a_8\) in the geometric progression \(15,45,135,\ldots\)?
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A (32805)
B (36450)
C (10935)
D (98415)
Explanation opens after your attempt
Correct Answer
A. (32805)
Step 1
Concept
\(a_8=15\cdot3^7=32805\). In exams, use \(r^7\) for the eighth term.
Step 2
Why this answer is correct
The correct answer is A. (32805). \(a_8=15\cdot3^7=32805\). In exams, use \(r^7\) for the eighth term.
Step 3
Exam Tip
\(a_8=15\cdot3^7=32805\) है। परीक्षा में आठवें पद के लिए \(r^7\) लगाएँ।
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यदि (a=128) और \(r=\frac{1}{2}\) है, तो पहले (5) पदों का योग क्या है?
If (a=128) and \(r=\frac{1}{2}\), what is the sum of the first (5) terms?
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A (248)
B (250)
C (252)
D (256)
Explanation opens after your attempt
Step 1
Concept
The terms are (128,64,32,16,8), and the sum is (248). In exams, direct addition is also easy for small decreasing terms.
Step 2
Why this answer is correct
The correct answer is A. (248). The terms are (128,64,32,16,8), and the sum is (248). In exams, direct addition is also easy for small decreasing terms.
Step 3
Exam Tip
पद (128,64,32,16,8) हैं और योग (248) है। परीक्षा में छोटे घटते पदों को सीधे जोड़ना भी आसान है।
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गुणोत्तर श्रेणी में \(a_2=24\) और \(a_5=648\) है, तो \(a_1\) क्या है?
In a geometric progression, \(a_2=24\) and \(a_5=648\). What is \(a_1\)?
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A (6)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
From \(\frac{648}{24}=27=r^3\), (r=3) and \(a_1=24\div3=8\). In exams, find (r) first and then \(a_1\).
Step 2
Why this answer is correct
The correct answer is B. (8). From \(\frac{648}{24}=27=r^3\), (r=3) and \(a_1=24\div3=8\). In exams, find (r) first and then \(a_1\).
Step 3
Exam Tip
\(\frac{648}{24}=27=r^3\) से (r=3) और \(a_1=24\div3=8\) है। परीक्षा में पहले (r) निकालें फिर \(a_1\)।
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यदि \(a_n=4\cdot5^{n-1}\) है, तो \(a_4+a_5\) का मान क्या है?
If \(a_n=4\cdot5^{n-1}\), what is the value of \(a_4+a_5\)?
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A (2500)
B (3000)
C (3125)
D (3500)
Explanation opens after your attempt
Step 1
Concept
\(a_4=500\) and \(a_5=2500\), so the sum is (3000). In exams, find both terms separately and add.
Step 2
Why this answer is correct
The correct answer is B. (3000). \(a_4=500\) and \(a_5=2500\), so the sum is (3000). In exams, find both terms separately and add.
Step 3
Exam Tip
\(a_4=500\) और \(a_5=2500\), इसलिए योग (3000) है। परीक्षा में दोनों पद अलग निकालकर जोड़ें।
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गुणोत्तर श्रेणी \(18,36,72,\ldots\) में \(a_n=2304\) हो, तो (n) क्या है?
In the geometric progression \(18,36,72,\ldots\), if \(a_n=2304\), what is (n)?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
From \(18\cdot2^{n-1}=2304\), \(2^{n-1}=128=2^7\), so (n=8). In exams, factor first and compare powers.
Step 2
Why this answer is correct
The correct answer is B. (8). From \(18\cdot2^{n-1}=2304\), \(2^{n-1}=128=2^7\), so (n=8). In exams, factor first and compare powers.
Step 3
Exam Tip
\(18\cdot2^{n-1}=2304\) से \(2^{n-1}=128=2^7\), इसलिए (n=8) है। परीक्षा में गुणनखंड निकालकर घात तुलना करें।
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यदि \(a_1=8\) और \(a_4=216\) है तथा (r) धनात्मक है, तो \(a_5\) क्या होगा?
If \(a_1=8\) and \(a_4=216\), and (r) is positive, what is \(a_5\)?
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A (432)
B (486)
C (648)
D (729)
Explanation opens after your attempt
Step 1
Concept
From \(216=8r^3\), \(r^3=27\) and (r=3), so \(a_5=216\cdot3=648\). In exams, find the ratio first.
Step 2
Why this answer is correct
The correct answer is C. (648). From \(216=8r^3\), \(r^3=27\) and (r=3), so \(a_5=216\cdot3=648\). In exams, find the ratio first.
Step 3
Exam Tip
\(216=8r^3\) से \(r^3=27\) और (r=3), इसलिए \(a_5=216\cdot3=648\) है। परीक्षा में पहले अनुपात निकालें।
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गुणोत्तर श्रेणी \(20,40,80,\ldots\) के पहले (8) पदों का योग क्या है?
What is the sum of the first (8) terms of the geometric progression \(20,40,80,\ldots\)?
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A (5100)
B (5120)
C (5140)
D (5160)
Explanation opens after your attempt
Step 1
Concept
(S_8=20\(2^8-1\)=5100). In exams, use \(2^8=256\).
Step 2
Why this answer is correct
The correct answer is A. (5100). (S_8=20\(2^8-1\)=5100). In exams, use \(2^8=256\).
Step 3
Exam Tip
(S_8=20\(2^8-1\)=5100) है। परीक्षा में \(2^8=256\) का उपयोग करें।
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यदि \(a_8=1792\) और (a=7) है तथा (r) धनात्मक है, तो (r) क्या होगा?
If \(a_8=1792\) and (a=7), and (r) is positive, what is (r)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
\(1792=7r^7\) gives \(r^7=256\), which does not match any option exactly. In exams, check (n-1) and powers carefully.
Step 2
Why this answer is correct
The correct answer is A. (2). \(1792=7r^7\) gives \(r^7=256\), which does not match any option exactly. In exams, check (n-1) and powers carefully.
Step 3
Exam Tip
\(1792=7r^7\) से \(r^7=256=2^8\) नहीं बनता, इसलिए दिए विकल्पों में कोई सही नहीं है। परीक्षा में (n-1) और घात दोनों ध्यान से जाँचें।
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गुणोत्तर श्रेणी \(150,50,\frac{50}{3},\frac{50}{9},\ldots\) में \(\frac{50}{81}\) कौन-सा पद है?
In the geometric progression \(150,50,\frac{50}{3},\frac{50}{9},\ldots\), which term is \(\frac{50}{81}\)?
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A छठा पद / (6)th term
B सातवाँ पद / (7)th term
C आठवाँ पद / (8)th term
D नौवाँ पद / (9)th term
Explanation opens after your attempt
Correct Answer
A. छठा पद / (6)th term
Step 1
Concept
Each term is multiplied by \(\frac{1}{3}\), and the sixth term is \(\frac{50}{81}\). In exams, fractional terms can also be checked in order.
Step 2
Why this answer is correct
The correct answer is A. छठा पद / (6)th term. Each term is multiplied by \(\frac{1}{3}\), and the sixth term is \(\frac{50}{81}\). In exams, fractional terms can also be checked in order.
Step 3
Exam Tip
हर बार \(\frac{1}{3}\) से गुणा होता है और छठा पद \(\frac{50}{81}\) आता है। परीक्षा में भिन्न पदों को क्रम से भी जाँच सकते हैं।
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यदि (a=5) और (r=4) है, तो \(a_5+a_6\) का मान क्या होगा?
If (a=5) and (r=4), what is the value of \(a_5+a_6\)?
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A (5120)
B (6400)
C (7680)
D (10240)
Explanation opens after your attempt
Step 1
Concept
\(a_5=5\cdot4^4=1280\) and \(a_6=5120\), so the sum is (6400). In exams, calculate large powers separately.
Step 2
Why this answer is correct
The correct answer is B. (6400). \(a_5=5\cdot4^4=1280\) and \(a_6=5120\), so the sum is (6400). In exams, calculate large powers separately.
Step 3
Exam Tip
\(a_5=5\cdot4^4=1280\) और \(a_6=5120\), इसलिए योग (6400) है। परीक्षा में बड़े घातों को अलग-अलग निकालें।
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गुणोत्तर श्रेणी \(6,18,54,\ldots\) में कितने पदों तक योग (2184) होगा?
How many terms of the geometric progression \(6,18,54,\ldots\) have sum (2184)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
(S_n=\frac{6\(3^n-1\)}{3-1}=3\(3^n-1\)), and (3\(3^n-1\)=2184) gives (n=6). In exams, simplify the sum and identify the power.
Step 2
Why this answer is correct
The correct answer is C. (6). (S_n=\frac{6\(3^n-1\)}{3-1}=3\(3^n-1\)), and (3\(3^n-1\)=2184) gives (n=6). In exams, simplify the sum and identify the power.
Step 3
Exam Tip
(S_n=\frac{6\(3^n-1\)}{3-1}=3\(3^n-1\)) और (3\(3^n-1\)=2184) से (n=6) है। परीक्षा में योग को सरल करके घात पहचानें।
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