गुणोत्तर श्रेणी \(3,6,12,24,\ldots\) का बारहवाँ पद क्या है?
What is the twelfth term of the geometric progression \(3,6,12,24,\ldots\)?
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A (3072)
B (6144)
C (8192)
D (12288)
Explanation opens after your attempt
Step 1
Concept
Here (a=3) and (r=2) so \(a_{12}=3\cdot2^{11}=6144\). In exams use \(a_n=ar^{n-1}\).
Step 2
Why this answer is correct
The correct answer is B. (6144). Here (a=3) and (r=2) so \(a_{12}=3\cdot2^{11}=6144\). In exams use \(a_n=ar^{n-1}\).
Step 3
Exam Tip
यहाँ (a=3) और (r=2) है इसलिए \(a_{12}=3\cdot2^{11}=6144\) है। परीक्षा में \(a_n=ar^{n-1}\) लगाएँ।
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यदि किसी गुणोत्तर श्रेणी में (a=5) और (r=3) है तो \(a_8\) का मान क्या होगा?
If a geometric progression has (a=5) and (r=3) what is the value of \(a_8\)?
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A (3645)
B (7290)
C (10935)
D (12150)
Explanation opens after your attempt
Correct Answer
C. (10935)
Step 1
Concept
\(a_8=5\cdot3^7=10935\). In exams use \(r^7\) for the eighth term.
Step 2
Why this answer is correct
The correct answer is C. (10935). \(a_8=5\cdot3^7=10935\). In exams use \(r^7\) for the eighth term.
Step 3
Exam Tip
\(a_8=5\cdot3^7=10935\) है। परीक्षा में आठवें पद के लिए \(r^7\) लगाएँ।
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गुणोत्तर श्रेणी \(256,128,64,32,\ldots\) का नौवाँ पद क्या है?
What is the ninth term of the geometric progression \(256,128,64,32,\ldots\)?
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A (1)
B (2)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
Here \(r=\frac{1}{2}\) so (a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1). In exams use the fractional ratio in decreasing progressions.
Step 2
Why this answer is correct
The correct answer is A. (1). Here \(r=\frac{1}{2}\) so (a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1). In exams use the fractional ratio in decreasing progressions.
Step 3
Exam Tip
यहाँ \(r=\frac{1}{2}\) है इसलिए (a_9=256\cdot\left\(\frac{1}{2}\right\)8 =1) है। परीक्षा में घटती श्रेणी में भिन्न अनुपात लगाएँ।
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यदि \(a_n=4\cdot3^{n-1}\) है तो (2916) कौन-सा पद होगा?
If \(a_n=4\cdot3^{n-1}\) which term will be (2916)?
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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
B. सातवाँ पद / (7)th term
Step 1
Concept
From \(4\cdot3^{n-1}=2916\), \(3^{n-1}=729=3^6\) so (n=7). In exams first equate the given term to \(a_n\).
Step 2
Why this answer is correct
The correct answer is B. सातवाँ पद / (7)th term. From \(4\cdot3^{n-1}=2916\), \(3^{n-1}=729=3^6\) so (n=7). In exams first equate the given term to \(a_n\).
Step 3
Exam Tip
\(4\cdot3^{n-1}=2916\) से \(3^{n-1}=729=3^6\) इसलिए (n=7) है। परीक्षा में पहले दिए पद को \(a_n\) के बराबर रखें।
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किसी गुणोत्तर श्रेणी में पहला पद (6) और पाँचवाँ पद (486) है। यदि (r) धनात्मक है तो (r) क्या है?
In a geometric progression the first term is (6) and the fifth term is (486). 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 \(486=6r^4\), \(r^4=81\) and (r=3). In exams remember \(r^4\) for the fifth term.
Step 2
Why this answer is correct
The correct answer is B. (3). From \(486=6r^4\), \(r^4=81\) and (r=3). In exams remember \(r^4\) for the fifth term.
Step 3
Exam Tip
\(486=6r^4\) से \(r^4=81\) और (r=3) है। परीक्षा में पाँचवें पद के लिए \(r^4\) याद रखें।
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गुणोत्तर श्रेणी \(10,30,90,270,\ldots\) का सामान्य पद क्या है?
What is the general term of the geometric progression \(10,30,90,270,\ldots\)?
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A \(a_n=10\cdot3^{n-1}\)
B \(a_n=3\cdot10^{n-1}\)
C \(a_n=10n+20\)
D \(a_n=30n\)
Explanation opens after your attempt
Correct Answer
A. \(a_n=10\cdot3^{n-1}\)
Step 1
Concept
The first term is (10) and the ratio is (3) so \(a_n=10\cdot3^{n-1}\). In exams keep (a) and (r) correct in \(ar^{n-1}\).
Step 2
Why this answer is correct
The correct answer is A. \(a_n=10\cdot3^{n-1}\). The first term is (10) and the ratio is (3) so \(a_n=10\cdot3^{n-1}\). In exams keep (a) and (r) correct in \(ar^{n-1}\).
Step 3
Exam Tip
पहला पद (10) और अनुपात (3) है इसलिए \(a_n=10\cdot3^{n-1}\) है। परीक्षा में \(ar^{n-1}\) में (a) और (r) सही रखें।
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गुणोत्तर श्रेणी \(2,6,18,54,\ldots\) के पहले (6) पदों का योग क्या है?
What is the sum of the first (6) terms of the geometric progression \(2,6,18,54,\ldots\)?
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A (486)
B (628)
C (728)
D (730)
Explanation opens after your attempt
Step 1
Concept
The sum of the first (6) terms is (2\(3^6-1\)/(3-1)=728). In exams use (S_n=\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 2
Why this answer is correct
The correct answer is C. (728). The sum of the first (6) terms is (2\(3^6-1\)/(3-1)=728). In exams use (S_n=\frac{a\(r^n-1\)}{r-1}) for (r>1).
Step 3
Exam Tip
पहले (6) पदों का योग (2\(3^6-1\)/(3-1)=728) है। परीक्षा में (r>1) के लिए (S_n=\frac{a\(r^n-1\)}{r-1}) लगाएँ।
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यदि (8,x,72) गुणोत्तर श्रेणी में हैं और (x>0) है तो (x) क्या होगा?
If (8,x,72) are in geometric progression and (x>0), what is (x)?
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A (18)
B (20)
C (24)
D (30)
Explanation opens after your attempt
Step 1
Concept
For the middle term \(x^2=8\cdot72=576\), so (x=24). In exams the square of the middle GP term equals the product of outer terms.
Step 2
Why this answer is correct
The correct answer is C. (24). For the middle term \(x^2=8\cdot72=576\), so (x=24). In exams the square of the middle GP term equals the product of outer terms.
Step 3
Exam Tip
मध्य पद के लिए \(x^2=8\cdot72=576\) इसलिए (x=24) 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 \(5,10,20,40,\ldots\) के पहले (8) पदों का योग क्या है?
What is the sum of the first (8) terms of the geometric progression \(5,10,20,40,\ldots\)?
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A (1275)
B (1280)
C (1295)
D (1305)
Explanation opens after your attempt
Step 1
Concept
(S_8=5\(2^8-1\)=1275). In exams calculate powers quickly when the ratio is (2).
Step 2
Why this answer is correct
The correct answer is A. (1275). (S_8=5\(2^8-1\)=1275). In exams calculate powers quickly when the ratio is (2).
Step 3
Exam Tip
(S_8=5\(2^8-1\)=1275) है। परीक्षा में अनुपात (2) हो तो घात जल्दी निकालें।
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यदि \(a_4=135\) और (r=3) है तो पहला पद (a) क्या होगा?
If \(a_4=135\) and (r=3), what is the first term (a)?
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A (3)
B (5)
C (9)
D (15)
Explanation opens after your attempt
Step 1
Concept
\(a_4=ar^3\), so \(135=a\cdot27\) and (a=5). In exams divide the fourth term by \(r^3\).
Step 2
Why this answer is correct
The correct answer is B. (5). \(a_4=ar^3\), so \(135=a\cdot27\) and (a=5). In exams divide the fourth term by \(r^3\).
Step 3
Exam Tip
\(a_4=ar^3\) इसलिए \(135=a\cdot27\) और (a=5) है। परीक्षा में चौथे पद से \(r^3\) भाग दें।
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गुणोत्तर श्रेणी \(81,27,9,3,\ldots\) का सामान्य पद क्या है?
What is the general term of the geometric progression \(81,27,9,3,\ldots\)?
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A \(a_n=81\cdot3^{n-1}\)
B (a_n=81\cdot\left\(\frac{1}{3}\right\)^{n-1})
C \(a_n=81-3n\)
D \(a_n=27n\)
Explanation opens after your attempt
Correct Answer
B. (a_n=81\cdot\left\(\frac{1}{3}\right\)^{n-1})
Step 1
Concept
The first term is (81) and the ratio is \(\frac{1}{3}\), so the correct rule is (81\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=81\cdot\left\(\frac{1}{3}\right\)^{n-1}). The first term is (81) and the ratio is \(\frac{1}{3}\), so the correct rule is (81\cdot\left\(\frac{1}{3}\right\)^{n-1}). In exams write the fractional ratio in a decreasing GP.
Step 3
Exam Tip
पहला पद (81) और अनुपात \(\frac{1}{3}\) है इसलिए सही नियम (81\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=4) और \(a_6=972\) है। यदि (r) धनात्मक है तो (r) क्या है?
For a geometric progression with (a=4) and \(a_6=972\), 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 \(972=4r^5\), \(r^5=243=3^5\) so (r=3). In exams \(r^5\) is used for the sixth term.
Step 2
Why this answer is correct
The correct answer is B. (3). From \(972=4r^5\), \(r^5=243=3^5\) so (r=3). In exams \(r^5\) is used for the sixth term.
Step 3
Exam Tip
\(972=4r^5\) से \(r^5=243=3^5\) इसलिए (r=3) है। परीक्षा में छठे पद के लिए \(r^5\) लगता है।
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गुणोत्तर श्रेणी \(3,12,48,192,\ldots\) में (3072) कौन-सा पद है?
In the geometric progression \(3,12,48,192,\ldots\), which term is (3072)?
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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
From \(3\cdot4^{n-1}=3072\), \(4^{n-1}=1024=4^5\) so (n=6). In exams equate powers to find (n).
Step 2
Why this answer is correct
The correct answer is A. छठा पद / (6)th term. From \(3\cdot4^{n-1}=3072\), \(4^{n-1}=1024=4^5\) so (n=6). In exams equate powers to find (n).
Step 3
Exam Tip
\(3\cdot4^{n-1}=3072\) 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=96) और \(r=\frac{1}{2}\) है तो \(a_7\) का मान क्या होगा?
If (a=96) and \(r=\frac{1}{2}\), what is the value of \(a_7\)?
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A (1)
B \(\frac{3}{2}\)
C (2)
D (3)
Explanation opens after your attempt
Correct Answer
B. \(\frac{3}{2}\)
Step 1
Concept
(a_7=96\cdot\left\(\frac{1}{2}\right\)6 =\frac{3}{2}). In exams apply powers of fractional ratios carefully.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{3}{2}\). (a_7=96\cdot\left\(\frac{1}{2}\right\)6 =\frac{3}{2}). In exams apply powers of fractional ratios carefully.
Step 3
Exam Tip
(a_7=96\cdot\left\(\frac{1}{2}\right\)6 =\frac{3}{2}) है। परीक्षा में भिन्न अनुपात के घात सावधानी से लगाएँ।
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यदि गुणोत्तर श्रेणी के पहले (5) पदों का योग (242) है और (a=2) तथा (r=3) है तो कथन कैसा है?
If the sum of the first (5) terms of a geometric progression is (242) with (a=2) and (r=3), what is the statement?
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A सही क्योंकि \(S_5=242\) / True because \(S_5=242\)
B गलत क्योंकि \(S_5=240\) / False because \(S_5=240\)
C गलत क्योंकि \(S_5=244\) / False because \(S_5=244\)
D सही क्योंकि \(S_5=121\) / True because \(S_5=121\)
Explanation opens after your attempt
Correct Answer
A. सही क्योंकि \(S_5=242\) / True because \(S_5=242\)
Step 1
Concept
(S_5=2\(3^5-1\)/(3-1)=242), so the statement is true. In exams verify statements by using the formula.
Step 2
Why this answer is correct
The correct answer is A. सही क्योंकि \(S_5=242\) / True because \(S_5=242\). (S_5=2\(3^5-1\)/(3-1)=242), so the statement is true. In exams verify statements by using the formula.
Step 3
Exam Tip
(S_5=2\(3^5-1\)/(3-1)=242) है इसलिए कथन सही है। परीक्षा में सूत्र लगाकर कथन की जाँच करें।
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गुणोत्तर श्रेणी \(1,3,9,27,\ldots\) के पहले (7) पदों का योग क्या है?
What is the sum of the first (7) terms of the geometric progression \(1,3,9,27,\ldots\)?
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A (1093)
B (1092)
C (1090)
D (729)
Explanation opens after your attempt
Step 1
Concept
\(S_7=\frac{3^7-1}{3-1}=1093\). In exams the form \(\frac{r^n-1}{r-1}\) is useful for (r>1).
Step 2
Why this answer is correct
The correct answer is A. (1093). \(S_7=\frac{3^7-1}{3-1}=1093\). In exams the form \(\frac{r^n-1}{r-1}\) is useful for (r>1).
Step 3
Exam Tip
\(S_7=\frac{3^7-1}{3-1}=1093\) है। परीक्षा में (r>1) के लिए \(\frac{r^n-1}{r-1}\) रूप उपयोगी है।
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गुणोत्तर श्रेणी में \(a_3=20\) और \(a_6=160\) है। यदि (r) धनात्मक है तो (r) क्या है?
In a geometric progression \(a_3=20\) and \(a_6=160\). If (r) is positive what is (r)?
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(\frac{a_6}{a_3}=r^3=\frac{160}{20}=8\), so (r=2). In exams the ratio of distant terms gives a power based on position gap.
Step 2
Why this answer is correct
The correct answer is A. (2). \(\frac{a_6}{a_3}=r^3=\frac{160}{20}=8\), so (r=2). In exams the ratio of distant terms gives a power based on position gap.
Step 3
Exam Tip
\(\frac{a_6}{a_3}=r^3=\frac{160}{20}=8\) इसलिए (r=2) है। परीक्षा में दूर के पदों का अनुपात पद-अंतर की घात देता है।
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यदि \(a_5=486\) और (r=3) है तो \(a_1\) का मान क्या है?
If \(a_5=486\) and (r=3), what is the value of \(a_1\)?
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A (2)
B (4)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(a_5=a\cdot3^4\), so \(a=486\div81=6\). In exams divide the fifth term by \(r^4\).
Step 2
Why this answer is correct
The correct answer is C. (6). \(a_5=a\cdot3^4\), so \(a=486\div81=6\). In exams divide the fifth term by \(r^4\).
Step 3
Exam Tip
\(a_5=a\cdot3^4\) इसलिए \(a=486\div81=6\) है। परीक्षा में पाँचवें पद से \(r^4\) भाग दें।
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गुणोत्तर श्रेणी \(5,15,45,135,\ldots\) का \(a_8\div a_5\) क्या होगा?
What is \(a_8\div a_5\) for the geometric progression \(5,15,45,135,\ldots\)?
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A (9)
B (18)
C (27)
D (81)
Explanation opens after your attempt
Step 1
Concept
The position gap is (3) and (r=3), so the ratio is \(3^3=27\). In exams use \(\frac{a_m}{a_n}=r^{m-n}\).
Step 2
Why this answer is correct
The correct answer is C. (27). The position gap is (3) and (r=3), so the ratio is \(3^3=27\). In exams use \(\frac{a_m}{a_n}=r^{m-n}\).
Step 3
Exam Tip
पद-संख्या का अंतर (3) है और (r=3) इसलिए अनुपात \(3^3=27\) है। परीक्षा में \(\frac{a_m}{a_n}=r^{m-n}\) लगाएँ।
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यदि \(a_n=7\cdot2^{n-1}\) है तो पहले (6) पदों का योग क्या होगा?
If \(a_n=7\cdot2^{n-1}\), what is the sum of the first (6) terms?
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A (420)
B (434)
C (441)
D (448)
Explanation opens after your attempt
Step 1
Concept
Here (a=7) and (r=2), so (S_6=7\(2^6-1\)=441). In exams identify (a) and (r) from the general term.
Step 2
Why this answer is correct
The correct answer is C. (441). Here (a=7) and (r=2), so (S_6=7\(2^6-1\)=441). In exams identify (a) and (r) from the general term.
Step 3
Exam Tip
यहाँ (a=7) और (r=2) है इसलिए (S_6=7\(2^6-1\)=441) है। परीक्षा में सामान्य पद से (a) और (r) पहचानें।
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गुणोत्तर श्रेणी \(2,10,50,\ldots\) का वह पद कौन-सा है जो (1250) है?
Which term of the geometric progression \(2,10,50,\ldots\) is (1250)?
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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 \(2\cdot5^{n-1}=1250\), \(5^{n-1}=625=5^4\) so (n=5). In exams compare powers to find the term number.
Step 2
Why this answer is correct
The correct answer is B. पाँचवाँ पद / (5)th term. From \(2\cdot5^{n-1}=1250\), \(5^{n-1}=625=5^4\) so (n=5). In exams compare powers to find the term number.
Step 3
Exam Tip
\(2\cdot5^{n-1}=1250\) से \(5^{n-1}=625=5^4\) इसलिए (n=5) है। परीक्षा में पद संख्या के लिए घात की तुलना करें।
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यदि \(a_2=12\) और \(a_5=324\) है तो (r) का धनात्मक मान क्या है?
If \(a_2=12\) and \(a_5=324\), 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
\(\frac{a_5}{a_2}=r^3=\frac{324}{12}=27\), so (r=3). In exams a gap of (3) positions gives \(r^3\).
Step 2
Why this answer is correct
The correct answer is B. (3). \(\frac{a_5}{a_2}=r^3=\frac{324}{12}=27\), so (r=3). In exams a gap of (3) positions gives \(r^3\).
Step 3
Exam Tip
\(\frac{a_5}{a_2}=r^3=\frac{324}{12}=27\) इसलिए (r=3) है। परीक्षा में पद-अंतर (3) हो तो \(r^3\) बनेगा।
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गुणोत्तर श्रेणी \(729,243,81,27,\ldots\) में (1) कौन-सा पद है?
In the geometric progression \(729,243,81,27,\ldots\), which term is (1)?
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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
A. सातवाँ पद / (7)th term
Step 1
Concept
From (729\left\(\frac{1}{3}\right\)^{n-1}=1), \(3^{6-(n-1)}=1\), so (n=7). In exams count decreasing powers carefully.
Step 2
Why this answer is correct
The correct answer is A. सातवाँ पद / (7)th term. From (729\left\(\frac{1}{3}\right\)^{n-1}=1), \(3^{6-(n-1)}=1\), so (n=7). In exams count decreasing powers carefully.
Step 3
Exam Tip
(729\left\(\frac{1}{3}\right\)^{n-1}=1) से \(3^{6-(n-1)}=1\) इसलिए (n=7) है। परीक्षा में घटती घातों को सावधानी से गिनें।
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यदि (a=3) और \(a_7=192\) है तथा (r) धनात्मक है तो (r) क्या होगा?
If (a=3) and \(a_7=192\), and (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 \(192=3r^6\), \(r^6=64=2^6\) so (r=2). In exams use \(r^6\) for the seventh term.
Step 2
Why this answer is correct
The correct answer is A. (2). From \(192=3r^6\), \(r^6=64=2^6\) so (r=2). In exams use \(r^6\) for the seventh term.
Step 3
Exam Tip
\(192=3r^6\) से \(r^6=64=2^6\) इसलिए (r=2) है। परीक्षा में सातवें पद के लिए \(r^6\) लगाएँ।
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गुणोत्तर श्रेणी \(4,12,36,\ldots\) के पहले (7) पदों का योग क्या है?
What is the sum of the first (7) terms of the geometric progression \(4,12,36,\ldots\)?
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A (4372)
B (4374)
C (4376)
D (4380)
Explanation opens after your attempt
Step 1
Concept
(S_7=4\(3^7-1\)/(3-1)=4372). In exams first calculate the value of \(3^7\).
Step 2
Why this answer is correct
The correct answer is A. (4372). (S_7=4\(3^7-1\)/(3-1)=4372). In exams first calculate the value of \(3^7\).
Step 3
Exam Tip
(S_7=4\(3^7-1\)/(3-1)=4372) है। परीक्षा में पहले \(3^7\) का मान निकालें।
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किस गुणोत्तर श्रेणी में \(a_3=48\) और \(a_6=384\) है। यदि (r) धनात्मक है तो \(a_1\) क्या है?
In a geometric progression \(a_3=48\) and \(a_6=384\). If (r) is positive what is \(a_1\)?
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A (6)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(\frac{384}{48}=8=r^3\), so (r=2) and \(a_1=\frac{48}{2^2}=12\). In exams find (r) first and then the first term.
Step 2
Why this answer is correct
The correct answer is C. (12). \(\frac{384}{48}=8=r^3\), so (r=2) and \(a_1=\frac{48}{2^2}=12\). In exams find (r) first and then the first term.
Step 3
Exam Tip
\(\frac{384}{48}=8=r^3\) से (r=2) और \(a_1=\frac{48}{2^2}=12\) है। परीक्षा में पहले (r) निकालें फिर पहला पद।
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यदि गुणोत्तर श्रेणी \(x,,2x,,4x,\ldots\) में पाँचवाँ पद (80) है तो (x) क्या है?
If the fifth term of the geometric progression \(x,,2x,,4x,\ldots\) is (80), what is (x)?
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A (4)
B (5)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
The fifth term is \(x\cdot2^4=16x=80\), 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 B. (5). The fifth term is \(x\cdot2^4=16x=80\), so (x=5). In exams put the algebraic first term in \(ar^{n-1}\) too.
Step 3
Exam Tip
पाँचवाँ पद \(x\cdot2^4=16x=80\) है इसलिए (x=5) है। परीक्षा में बीजीय पहले पद को भी \(ar^{n-1}\) में रखें।
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गुणोत्तर श्रेणी \(1,4,16,\ldots\) में (1024) तक कितने पद हैं?
How many terms are there up to (1024) in the geometric progression \(1,4,16,\ldots\)?
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(1\cdot4^{n-1}=1024=4^5\), so (n=6). In exams equate the last term to the general term.
Step 2
Why this answer is correct
The correct answer is B. (6). \(1\cdot4^{n-1}=1024=4^5\), so (n=6). In exams equate the last term to the general term.
Step 3
Exam Tip
\(1\cdot4^{n-1}=1024=4^5\) इसलिए (n=6) है। परीक्षा में अंतिम पद को सामान्य पद के बराबर रखें।
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यदि किसी गुणोत्तर श्रेणी के पहले (4) पद (2,8,32,128) हैं तो \(S_4\) क्या है?
If the first (4) terms of a geometric progression are (2,8,32,128), what is \(S_4\)?
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A (160)
B (168)
C (170)
D (180)
Explanation opens after your attempt
Step 1
Concept
The sum is (2+8+32+128=170). In exams direct addition is also fast for small (n).
Step 2
Why this answer is correct
The correct answer is C. (170). The sum is (2+8+32+128=170). In exams direct addition is also fast for small (n).
Step 3
Exam Tip
योग (2+8+32+128=170) है। परीक्षा में छोटे (n) के लिए सीधे जोड़ना भी तेज होता है।
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गुणोत्तर श्रेणी \(6,24,96,\ldots\) का कौन-सा पद (6144) है?
Which term of the geometric progression \(6,24,96,\ldots\) is (6144)?
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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
From \(6\cdot4^{n-1}=6144\), \(4^{n-1}=1024=4^5\), so (n=6). In exams equate powers to find the term number.
Step 2
Why this answer is correct
The correct answer is A. छठा पद / (6)th term. From \(6\cdot4^{n-1}=6144\), \(4^{n-1}=1024=4^5\), so (n=6). In exams equate powers to find the term number.
Step 3
Exam Tip
\(6\cdot4^{n-1}=6144\) से \(4^{n-1}=1024=4^5\) इसलिए (n=6) है। परीक्षा में घात बराबर करके पद संख्या निकालें।
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यदि \(a_1=5\) और (r=-2) है तो \(a_6\) क्या होगा?
If \(a_1=5\) and (r=-2), what is \(a_6\)?
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A (-160)
B (160)
C (-80)
D (80)
Explanation opens after your attempt
Step 1
Concept
(a_6=5(-2)5 =-160). In exams an odd power of a negative ratio remains negative.
Step 2
Why this answer is correct
The correct answer is A. (-160). (a_6=5(-2)5 =-160). In exams an odd power of a negative ratio remains negative.
Step 3
Exam Tip
(a_6=5(-2)5 =-160) है। परीक्षा में ऋणात्मक अनुपात की विषम घात ऋणात्मक रहती है।
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गुणोत्तर श्रेणी \(3,15,75,375,\ldots\) के पहले (5) पदों का योग क्या है?
What is the sum of the first (5) terms of the geometric progression \(3,15,75,375,\ldots\)?
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A (2340)
B (2343)
C (2346)
D (2350)
Explanation opens after your attempt
Step 1
Concept
(S_5=3\(5^5-1\)/(5-1)=2343). In exams the formula saves time for large ratios.
Step 2
Why this answer is correct
The correct answer is B. (2343). (S_5=3\(5^5-1\)/(5-1)=2343). In exams the formula saves time for large ratios.
Step 3
Exam Tip
(S_5=3\(5^5-1\)/(5-1)=2343) है। परीक्षा में बड़े अनुपात में सूत्र से समय बचता है।
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यदि \(a_4=80\) और (r=2) है तो \(a_9\) का मान क्या होगा?
If \(a_4=80\) and (r=2), what is the value of \(a_9\)?
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A (1280)
B (1920)
C (2560)
D (5120)
Explanation opens after your attempt
Step 1
Concept
There are (5) steps from the fourth to the ninth term, so \(a_9=80\cdot2^5=2560\). In exams count the position gap.
Step 2
Why this answer is correct
The correct answer is C. (2560). There are (5) steps from the fourth to the ninth term, so \(a_9=80\cdot2^5=2560\). In exams count the position gap.
Step 3
Exam Tip
चौथे से नौवें पद तक (5) कदम हैं इसलिए \(a_9=80\cdot2^5=2560\) है। परीक्षा में पद-संख्या का अंतर गिनें।
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गुणोत्तर श्रेणी \(512,256,128,\ldots\) में (4) कौन-सा पद है?
In the geometric progression \(512,256,128,\ldots\), which term is (4)?
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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
From (512\left\(\frac{1}{2}\right\)^{n-1}=4), (\left\(\frac{1}{2}\right\)^{n-1}=\frac{1}{128}), so (n=8). In exams count terms while halving.
Step 2
Why this answer is correct
The correct answer is B. आठवाँ पद / (8)th term. From (512\left\(\frac{1}{2}\right\)^{n-1}=4), (\left\(\frac{1}{2}\right\)^{n-1}=\frac{1}{128}), so (n=8). In exams count terms while halving.
Step 3
Exam Tip
(512\left\(\frac{1}{2}\right\)^{n-1}=4) से (\left\(\frac{1}{2}\right\)^{n-1}=\frac{1}{128}) इसलिए (n=8) है। परीक्षा में आधा करते हुए पद गिनें।
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यदि (a=6) और \(S_4=240\) है तथा (r=3) है तो कथन कैसा है?
If (a=6), \(S_4=240\), and (r=3), what is the statement?
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A सही क्योंकि \(S_4=240\) / True because \(S_4=240\)
B गलत क्योंकि \(S_4=120\) / False because \(S_4=120\)
C गलत क्योंकि \(S_4=180\) / False because \(S_4=180\)
D गलत क्योंकि \(S_4=300\) / False because \(S_4=300\)
Explanation opens after your attempt
Correct Answer
A. सही क्योंकि \(S_4=240\) / True because \(S_4=240\)
Step 1
Concept
(S_4=6\(3^4-1\)/(3-1)=240). In exams confirm statement-type questions with the formula.
Step 2
Why this answer is correct
The correct answer is A. सही क्योंकि \(S_4=240\) / True because \(S_4=240\). (S_4=6\(3^4-1\)/(3-1)=240). In exams confirm statement-type questions with the formula.
Step 3
Exam Tip
(S_4=6\(3^4-1\)/(3-1)=240) है। परीक्षा में कथन-प्रकार प्रश्न में सूत्र से पुष्टि करें।
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गुणोत्तर श्रेणी \(8,24,72,\ldots\) के \(a_5+a_6\) का मान क्या है?
What is the value of \(a_5+a_6\) for the geometric progression \(8,24,72,\ldots\)?
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A (864)
B (972)
C (1080)
D (1296)
Explanation opens after your attempt
Step 1
Concept
\(a_5=648\) and \(a_6=1944\), so their sum is (2592). None of the given options is correct.
Step 2
Why this answer is correct
The correct answer is C. (1080). \(a_5=648\) and \(a_6=1944\), so their sum is (2592). None of the given options is correct.
Step 3
Exam Tip
\(a_5=648\) और \(a_6=1944\) नहीं बल्कि \(a_5=648\) तथा \(a_6=1944\) का योग (2592) होता है। दिए विकल्पों में सही उत्तर नहीं है।
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यदि \(a_n=100\left(\frac{1}{2}\right)^{n-1}\) है तो कौन-सा पद \(\frac{25}{2}\) के बराबर होगा?
If \(a_n=100\left(\frac{1}{2}\right)^{n-1}\), which term will be equal to \(\frac{25}{2}\)?
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A तीसरा पद / (3)rd term
B चौथा पद / (4)th term
C पाँचवाँ पद / (5)th term
D छठा पद / (6)th term
Explanation opens after your attempt
Correct Answer
B. चौथा पद / (4)th term
Step 1
Concept
From \(100\left(\frac{1}{2}\right)^{n-1}=\frac{25}{2}\), \(\left(\frac{1}{2}\right)^{n-1}=\frac{1}{8}\), so (n=4). In exams simplify fractions first.
Step 2
Why this answer is correct
The correct answer is B. चौथा पद / (4)th term. From \(100\left(\frac{1}{2}\right)^{n-1}=\frac{25}{2}\), \(\left(\frac{1}{2}\right)^{n-1}=\frac{1}{8}\), so (n=4). In exams simplify fractions first.
Step 3
Exam Tip
\(100\left(\frac{1}{2}\right)^{n-1}=\frac{25}{2}\) से \(\left(\frac{1}{2}\right)^{n-1}=\frac{1}{8}\) इसलिए (n=4) है। परीक्षा में भिन्नों को पहले सरल करें।
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गुणोत्तर श्रेणी \(4,8,16,\ldots\) के पहले (10) पदों का योग क्या है?
What is the sum of the first (10) terms of the geometric progression \(4,8,16,\ldots\)?
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A (4092)
B (4096)
C (2044)
D (2048)
Explanation opens after your attempt
Step 1
Concept
(S_{10}=4\(2^{10}-1\)=4092). In exams remembering \(2^{10}=1024\) is useful.
Step 2
Why this answer is correct
The correct answer is A. (4092). (S_{10}=4\(2^{10}-1\)=4092). In exams remembering \(2^{10}=1024\) is useful.
Step 3
Exam Tip
(S_{10}=4\(2^{10}-1\)=4092) है। परीक्षा में \(2^{10}=1024\) याद रखना उपयोगी है।
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यदि किसी गुणोत्तर श्रेणी में \(a_4=54\) और \(a_7=1458\) है तो (r) का धनात्मक मान क्या है?
If a geometric progression has \(a_4=54\) and \(a_7=1458\), 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
\(\frac{1458}{54}=27=r^3\), so (r=3). In exams a position gap of (3) gives \(r^3\).
Step 2
Why this answer is correct
The correct answer is B. (3). \(\frac{1458}{54}=27=r^3\), so (r=3). In exams a position gap of (3) gives \(r^3\).
Step 3
Exam Tip
\(\frac{1458}{54}=27=r^3\) इसलिए (r=3) है। परीक्षा में पद-संख्या का अंतर (3) होने पर \(r^3\) मिलेगा।
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गुणोत्तर श्रेणी \(6,12,24,\ldots\) में \(a_{11}\) क्या है?
What is \(a_{11}\) in the geometric progression \(6,12,24,\ldots\)?
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A (3072)
B (4096)
C (6144)
D (12288)
Explanation opens after your attempt
Step 1
Concept
\(a_{11}=6\cdot2^{10}=6144\). In exams use \(r^{10}\) for the eleventh term.
Step 2
Why this answer is correct
The correct answer is C. (6144). \(a_{11}=6\cdot2^{10}=6144\). In exams use \(r^{10}\) for the eleventh term.
Step 3
Exam Tip
\(a_{11}=6\cdot2^{10}=6144\) है। परीक्षा में ग्यारहवें पद के लिए \(r^{10}\) लगाएँ।
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यदि (a=81) और \(r=\frac{1}{3}\) है तो पहले (5) पदों का योग क्या है?
If (a=81) and \(r=\frac{1}{3}\), what is the sum of the first (5) terms?
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A (120)
B (121)
C (122)
D (123)
Explanation opens after your attempt
Step 1
Concept
The terms are (81,27,9,3,1) and the sum is (121). In exams direct addition is also easy for small decreasing terms.
Step 2
Why this answer is correct
The correct answer is B. (121). The terms are (81,27,9,3,1) and the sum is (121). In exams direct addition is also easy for small decreasing terms.
Step 3
Exam Tip
पद (81,27,9,3,1) हैं और योग (121) है। परीक्षा में छोटे घटते पदों को सीधे जोड़ना भी आसान है।
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गुणोत्तर श्रेणी में \(a_2=18\) और \(a_5=486\) है तो \(a_1\) क्या है?
In a geometric progression \(a_2=18\) and \(a_5=486\). What is \(a_1\)?
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A (2)
B (4)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
From \(\frac{486}{18}=27=r^3\), (r=3) and \(a_1=18\div3=6\). In exams find (r) first and then \(a_1\).
Step 2
Why this answer is correct
The correct answer is C. (6). From \(\frac{486}{18}=27=r^3\), (r=3) and \(a_1=18\div3=6\). In exams find (r) first and then \(a_1\).
Step 3
Exam Tip
\(\frac{486}{18}=27=r^3\) से (r=3) और \(a_1=18\div3=6\) है। परीक्षा में पहले (r) निकालें फिर \(a_1\)।
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यदि \(a_n=3\cdot4^{n-1}\) है तो \(a_4+a_5\) का मान क्या है?
If \(a_n=3\cdot4^{n-1}\), what is the value of \(a_4+a_5\)?
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A (960)
B (1020)
C (1080)
D (1200)
Explanation opens after your attempt
Step 1
Concept
\(a_4=192\) and \(a_5=768\), so the sum is (960). In exams find both terms separately and add.
Step 2
Why this answer is correct
The correct answer is A. (960). \(a_4=192\) and \(a_5=768\), so the sum is (960). In exams find both terms separately and add.
Step 3
Exam Tip
\(a_4=192\) और \(a_5=768\) इसलिए योग (960) है। परीक्षा में दोनों पद अलग निकालकर जोड़ें।
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गुणोत्तर श्रेणी \(11,22,44,\ldots\) में \(a_n=1408\) हो तो (n) क्या है?
In the geometric progression \(11,22,44,\ldots\), if \(a_n=1408\), what is (n)?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
From \(11\cdot2^{n-1}=1408\), \(2^{n-1}=128=2^7\), so (n=8). In exams factor first and compare powers.
Step 2
Why this answer is correct
The correct answer is B. (8). From \(11\cdot2^{n-1}=1408\), \(2^{n-1}=128=2^7\), so (n=8). In exams factor first and compare powers.
Step 3
Exam Tip
\(11\cdot2^{n-1}=1408\) से \(2^{n-1}=128=2^7\) इसलिए (n=8) है। परीक्षा में गुणनखंड निकालकर घात तुलना करें।
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यदि \(a_1=7\) और \(a_4=189\) है तथा (r) धनात्मक है तो \(a_6\) क्या होगा?
If \(a_1=7\) and \(a_4=189\), and (r) is positive, what is \(a_6\)?
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A (1701)
B (1530)
C (1458)
D (1323)
Explanation opens after your attempt
Step 1
Concept
From \(189=7r^3\), (r=3) and \(a_6=7\cdot3^5=1701\). In exams find the ratio first.
Step 2
Why this answer is correct
The correct answer is A. (1701). From \(189=7r^3\), (r=3) and \(a_6=7\cdot3^5=1701\). In exams find the ratio first.
Step 3
Exam Tip
\(189=7r^3\) से (r=3) और \(a_6=7\cdot3^5=1701\) है। परीक्षा में पहले अनुपात निकालें।
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गुणोत्तर श्रेणी \(9,18,36,\ldots\) के पहले (12) पदों का योग क्या है?
What is the sum of the first (12) terms of the geometric progression \(9,18,36,\ldots\)?
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A (36855)
B (36864)
C (32760)
D (40950)
Explanation opens after your attempt
Correct Answer
A. (36855)
Step 1
Concept
(S_{12}=9\(2^{12}-1\)=36855). In exams use \(2^{12}=4096\).
Step 2
Why this answer is correct
The correct answer is A. (36855). (S_{12}=9\(2^{12}-1\)=36855). In exams use \(2^{12}=4096\).
Step 3
Exam Tip
(S_{12}=9\(2^{12}-1\)=36855) है। परीक्षा में \(2^{12}=4096\) का उपयोग करें।
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यदि \(a_6=224\) और (a=7) है तथा (r) धनात्मक है तो (r) क्या होगा?
If \(a_6=224\) 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
From \(224=7r^5\), \(r^5=32=2^5\), so (r=2). In exams identify \(r^5\) for the sixth term.
Step 2
Why this answer is correct
The correct answer is A. (2). From \(224=7r^5\), \(r^5=32=2^5\), so (r=2). In exams identify \(r^5\) for the sixth term.
Step 3
Exam Tip
\(224=7r^5\) से \(r^5=32=2^5\) इसलिए (r=2) है। परीक्षा में छठे पद के लिए \(r^5\) पहचानें।
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गुणोत्तर श्रेणी \(45,15,5,\frac{5}{3},\ldots\) में \(\frac{5}{27}\) कौन-सा पद है?
In the geometric progression \(45,15,5,\frac{5}{3},\ldots\), which term is \(\frac{5}{27}\)?
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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 \(\frac{5}{27}\). 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 \(\frac{5}{27}\). 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 \(\frac{5}{27}\) आता है। परीक्षा में भिन्न पदों को क्रम से भी जाँच सकते हैं।
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यदि (a=2) और (r=4) है तो \(a_5+a_6\) का मान क्या होगा?
If (a=2) and (r=4), what is the value of \(a_5+a_6\)?
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A (2048)
B (2560)
C (3072)
D (4096)
Explanation opens after your attempt
Step 1
Concept
\(a_5=2\cdot4^4=512\) and \(a_6=2048\), so the sum is (2560). In exams calculate large powers separately.
Step 2
Why this answer is correct
The correct answer is B. (2560). \(a_5=2\cdot4^4=512\) and \(a_6=2048\), so the sum is (2560). In exams calculate large powers separately.
Step 3
Exam Tip
\(a_5=2\cdot4^4=512\) और \(a_6=2048\) इसलिए योग (2560) है। परीक्षा में बड़े घातों को अलग-अलग निकालें।
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गुणोत्तर श्रेणी \(2,6,18,\ldots\) में कितने पदों तक योग (242) होगा?
How many terms of the geometric progression \(2,6,18,\ldots\) have sum (242)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
(S_n=2\(3^n-1\)/(3-1)=3^n-1), and \(3^n-1=242\) gives (n=5). In exams simplify the sum and identify the power.
Step 2
Why this answer is correct
The correct answer is B. (5). (S_n=2\(3^n-1\)/(3-1)=3^n-1), and \(3^n-1=242\) gives (n=5). In exams simplify the sum and identify the power.
Step 3
Exam Tip
(S_n=2\(3^n-1\)/(3-1)=3^n-1) और \(3^n-1=242\) से (n=5) है। परीक्षा में योग को सरल करके घात पहचानें।
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