अनुक्रम \(3,6,12,24,\ldots\) में सार्व अनुपात क्या है?
What is the common ratio in the sequence \(3,6,12,24,\ldots\)?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The common ratio is \(\frac{6}{3}=2\). In a geometric progression divide the second term by the first term.
Step 2
Why this answer is correct
The correct answer is A. (2). The common ratio is \(\frac{6}{3}=2\). In a geometric progression divide the second term by the first term.
Step 3
Exam Tip
सार्व अनुपात \(\frac{6}{3}=2\) है। गुणोत्तर श्रेढ़ी में दूसरा पद पहले पद से भाग दें।
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यदि किसी गुणोत्तर श्रेढ़ी में पहला पद (5) और सार्व अनुपात (3) है, तो चौथा पद क्या होगा?
If a geometric progression has first term (5) and common ratio (3), what is the fourth term?
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#nth-term
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A (45)
B (90)
C (135)
D (150)
Explanation opens after your attempt
Step 1
Concept
\(a_4=5\cdot3^3=135\). Remember \(a_n=ar^{n-1}\) for the (n)th term.
Step 2
Why this answer is correct
The correct answer is C. (135). \(a_4=5\cdot3^3=135\). Remember \(a_n=ar^{n-1}\) for the (n)th term.
Step 3
Exam Tip
\(a_4=5\cdot3^3=135\)। (n)वें पद के लिए \(a_n=ar^{n-1}\) याद रखें।
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कौन-सा अनुक्रम गुणोत्तर श्रेढ़ी है?
Which sequence is a geometric progression?
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A \(2,5,8,11,\ldots\)
B \(4,12,36,108,\ldots\)
C \(1,4,9,16,\ldots\)
D \(6,10,14,18,\ldots\)
Explanation opens after your attempt
Correct Answer
B. \(4,12,36,108,\ldots\)
Step 1
Concept
In \(4,12,36,108,\ldots\), each term is multiplied by (3). In a geometric progression consecutive ratios are equal.
Step 2
Why this answer is correct
The correct answer is B. \(4,12,36,108,\ldots\). In \(4,12,36,108,\ldots\), each term is multiplied by (3). In a geometric progression consecutive ratios are equal.
Step 3
Exam Tip
\(4,12,36,108,\ldots\) में हर बार (3) से गुणा हो रहा है। गुणोत्तर श्रेढ़ी में लगातार अनुपात समान होता है।
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अनुक्रम \(81,27,9,3,\ldots\) का सार्व अनुपात क्या है?
What is the common ratio of the sequence \(81,27,9,3,\ldots\)?
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A (3)
B \(\frac{1}{3}\)
C \(\frac{1}{9}\)
D (9)
Explanation opens after your attempt
Correct Answer
B. \(\frac{1}{3}\)
Step 1
Concept
The common ratio is \(\frac{27}{81}=\frac{1}{3}\). In a decreasing geometric progression the ratio can be less than (1).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{1}{3}\). The common ratio is \(\frac{27}{81}=\frac{1}{3}\). In a decreasing geometric progression the ratio can be less than (1).
Step 3
Exam Tip
सार्व अनुपात \(\frac{27}{81}=\frac{1}{3}\) है। घटती गुणोत्तर श्रेढ़ी में अनुपात (1) से छोटा हो सकता है।
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यदि \(a_n=2\cdot4^{n-1}\) है, तो \(a_5\) का मान क्या होगा?
If \(a_n=2\cdot4^{n-1}\), what is the value of \(a_5\)?
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A (128)
B (256)
C (512)
D (1024)
Explanation opens after your attempt
Step 1
Concept
\(a_5=2\cdot4^4=512\). First find the value of (n-1) in the exponent.
Step 2
Why this answer is correct
The correct answer is C. (512). \(a_5=2\cdot4^4=512\). First find the value of (n-1) in the exponent.
Step 3
Exam Tip
\(a_5=2\cdot4^4=512\)। घात में (n-1) का मान पहले निकालें।
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गुणोत्तर श्रेढ़ी \(7,14,28,56,\ldots\) में (224) कौन-सा पद है?
In the geometric progression \(7,14,28,56,\ldots\), which term is (224)?
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A (4)वाँ / (4)th
B (5)वाँ / (5)th
C (6)वाँ / (6)th
D (7)वाँ / (7)th
Explanation opens after your attempt
Correct Answer
C. (6)वाँ / (6)th
Step 1
Concept
The terms are (7,14,28,56,112,224), so (224) is the sixth term. For small questions listing terms is quick.
Step 2
Why this answer is correct
The correct answer is C. (6)वाँ / (6)th. The terms are (7,14,28,56,112,224), so (224) is the sixth term. For small questions listing terms is quick.
Step 3
Exam Tip
पद (7,14,28,56,112,224) हैं, इसलिए (224) छठा पद है। छोटे प्रश्नों में पद लिखकर जांचना तेज है।
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कौन-सा अनुक्रम गुणोत्तर श्रेढ़ी नहीं है?
Which sequence is not a geometric progression?
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A \(5,10,20,40,\ldots\)
B \(3,9,27,81,\ldots\)
C \(2,4,8,15,\ldots\)
D \(16,8,4,2,\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(2,4,8,15,\ldots\)
Step 1
Concept
In \(2,4,8,15,\ldots\), the ratios \(2,2,\frac{15}{8}\) are not equal. A geometric progression must have a constant ratio.
Step 2
Why this answer is correct
The correct answer is C. \(2,4,8,15,\ldots\). In \(2,4,8,15,\ldots\), the ratios \(2,2,\frac{15}{8}\) are not equal. A geometric progression must have a constant ratio.
Step 3
Exam Tip
\(2,4,8,15,\ldots\) में अनुपात \(2,2,\frac{15}{8}\) बराबर नहीं हैं। गुणोत्तर श्रेढ़ी में अनुपात स्थिर होना चाहिए।
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यदि किसी गुणोत्तर श्रेढ़ी में (a=6) और (r=2) है, तो पहले चार पद कौन-से होंगे?
If a geometric progression has (a=6) and (r=2), what are the first four terms?
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A (6,8,10,12)
B (6,12,24,48)
C (2,6,18,54)
D (6,18,54,162)
Explanation opens after your attempt
Correct Answer
B. (6,12,24,48)
Step 1
Concept
The first term is (6), and multiplying by (2) each time gives (6,12,24,48). In a geometric progression each next term is formed by multiplication.
Step 2
Why this answer is correct
The correct answer is B. (6,12,24,48). The first term is (6), and multiplying by (2) each time gives (6,12,24,48). In a geometric progression each next term is formed by multiplication.
Step 3
Exam Tip
पहला पद (6) है और हर बार (2) से गुणा करने पर (6,12,24,48) मिलते हैं। गुणोत्तर श्रेढ़ी में अगला पद गुणा करके बनता है।
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अनुक्रम \(2,-6,18,-54,\ldots\) का सार्व अनुपात क्या है?
What is the common ratio of the sequence \(2,-6,18,-54,\ldots\)?
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A (3)
B (-3)
C (6)
D (-6)
Explanation opens after your attempt
Step 1
Concept
The common ratio is \(\frac{-6}{2}=-3\). With a negative ratio the signs alternate.
Step 2
Why this answer is correct
The correct answer is B. (-3). The common ratio is \(\frac{-6}{2}=-3\). With a negative ratio the signs alternate.
Step 3
Exam Tip
सार्व अनुपात \(\frac{-6}{2}=-3\) है। ऋणात्मक अनुपात होने पर चिह्न बारी-बारी से बदलते हैं।
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गुणोत्तर श्रेढ़ी \(4,12,36,\ldots\) में अगला पद क्या होगा?
What is the next term in the geometric progression \(4,12,36,\ldots\)?
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A (48)
B (72)
C (96)
D (108)
Explanation opens after your attempt
Step 1
Concept
The common ratio is (3), so the next term is \(36\cdot3=108\). Multiply the previous term by the ratio for the next term.
Step 2
Why this answer is correct
The correct answer is D. (108). The common ratio is (3), so the next term is \(36\cdot3=108\). Multiply the previous term by the ratio for the next term.
Step 3
Exam Tip
सार्व अनुपात (3) है, इसलिए अगला पद \(36\cdot3=108\) होगा। अगले पद के लिए पिछले पद को अनुपात से गुणा करें।
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यदि (3,x,27) गुणोत्तर श्रेढ़ी के लगातार तीन धनात्मक पद हैं, तो (x) क्या होगा?
If (3,x,27) are three consecutive positive terms of a geometric progression, what is (x)?
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A (6)
B (9)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
For the middle term (x), \(x^2=3\cdot27=81\), so (x=9). For positive terms take the positive geometric mean.
Step 2
Why this answer is correct
The correct answer is B. (9). For the middle term (x), \(x^2=3\cdot27=81\), so (x=9). For positive terms take the positive geometric mean.
Step 3
Exam Tip
मध्य पद (x) के लिए \(x^2=3\cdot27=81\), इसलिए (x=9)। धनात्मक पदों में ज्यामितीय माध्य धनात्मक लें।
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गुणोत्तर श्रेढ़ी \(5,15,45,135,\ldots\) के पहले (4) पदों का योग क्या है?
What is the sum of the first (4) terms of the geometric progression \(5,15,45,135,\ldots\)?
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A (180)
B (190)
C (200)
D (210)
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Step 1
Concept
The sum is (5+15+45+135=200). For a small number of terms direct addition is safe.
Step 2
Why this answer is correct
The correct answer is C. (200). The sum is (5+15+45+135=200). For a small number of terms direct addition is safe.
Step 3
Exam Tip
योग (5+15+45+135=200) है। कम पद हों तो सीधे जोड़ना सुरक्षित है।
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यदि किसी गुणोत्तर श्रेढ़ी में \(a_2=10\) और (r=2) है, तो \(a_1\) क्या होगा?
If a geometric progression has \(a_2=10\) and (r=2), what is \(a_1\)?
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A (2)
B (5)
C (10)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(a_2=a_1r\), so \(10=a_1\cdot2\) and \(a_1=5\). Divide by the ratio to move backward.
Step 2
Why this answer is correct
The correct answer is B. (5). \(a_2=a_1r\), so \(10=a_1\cdot2\) and \(a_1=5\). Divide by the ratio to move backward.
Step 3
Exam Tip
\(a_2=a_1r\), इसलिए \(10=a_1\cdot2\) और \(a_1=5\)। पीछे जाने के लिए अनुपात से भाग दें।
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यदि गुणोत्तर श्रेढ़ी में \(a_1=4\) और \(a_4=108\) है, तो सार्व अनुपात क्या है?
If a geometric progression has \(a_1=4\) and \(a_4=108\), what is the common ratio?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(a_4=a_1r^3\), so \(108=4r^3\) and \(r^3=27\), hence (r=3). Use the position gap as the exponent.
Step 2
Why this answer is correct
The correct answer is B. (3). \(a_4=a_1r^3\), so \(108=4r^3\) and \(r^3=27\), hence (r=3). Use the position gap as the exponent.
Step 3
Exam Tip
\(a_4=a_1r^3\), इसलिए \(108=4r^3\) और \(r^3=27\), अतः (r=3)। पद-संख्या के अंतर को घात बनाएं।
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अनुक्रम \(\frac{1}{2},1,2,4,\ldots\) का (6)वाँ पद क्या होगा?
What is the (6)th term of the sequence \(\frac{1}{2},1,2,4,\ldots\)?
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A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
The common ratio is (2), and the terms are \(\frac{1}{2},1,2,4,8,16\). Keep multiplying in order.
Step 2
Why this answer is correct
The correct answer is B. (16). The common ratio is (2), and the terms are \(\frac{1}{2},1,2,4,8,16\). Keep multiplying in order.
Step 3
Exam Tip
सार्व अनुपात (2) है और पद \(\frac{1}{2},1,2,4,8,16\) हैं। क्रम से गुणा करते जाएं।
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गुणोत्तर श्रेढ़ी \(64,32,16,8,\ldots\) का पाँचवाँ पद क्या है?
What is the fifth term of the geometric progression \(64,32,16,8,\ldots\)?
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The common ratio is \(\frac{1}{2}\), so the fifth term is (4). In a decreasing geometric progression multiply by the ratio repeatedly.
Step 2
Why this answer is correct
The correct answer is B. (4). The common ratio is \(\frac{1}{2}\), so the fifth term is (4). In a decreasing geometric progression multiply by the ratio repeatedly.
Step 3
Exam Tip
सार्व अनुपात \(\frac{1}{2}\) है, इसलिए पाँचवाँ पद (4) है। घटती गुणोत्तर श्रेढ़ी में बार-बार अनुपात से गुणा करें।
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यदि (a=3) और (r=5) है, तो \(a_4\) क्या होगा?
If (a=3) and (r=5), what is \(a_4\)?
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A (75)
B (125)
C (375)
D (625)
Explanation opens after your attempt
Step 1
Concept
\(a_4=3\cdot5^3=375\). In the (n)th term the exponent is (n-1).
Step 2
Why this answer is correct
The correct answer is C. (375). \(a_4=3\cdot5^3=375\). In the (n)th term the exponent is (n-1).
Step 3
Exam Tip
\(a_4=3\cdot5^3=375\)। (n)वें पद में घात (n-1) होती है।
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किस विकल्प में सार्व अनुपात \(\frac{1}{4}\) है?
Which option has common ratio \(\frac{1}{4}\)?
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A \(4,8,16,32,\ldots\)
B \(64,16,4,1,\ldots\)
C \(1,4,16,64,\ldots\)
D \(8,4,2,1,\ldots\)
Explanation opens after your attempt
Correct Answer
B. \(64,16,4,1,\ldots\)
Step 1
Concept
In \(64,16,4,1,\ldots\), \(\frac{16}{64}=\frac{1}{4}\). Divide consecutive terms to find the ratio.
Step 2
Why this answer is correct
The correct answer is B. \(64,16,4,1,\ldots\). In \(64,16,4,1,\ldots\), \(\frac{16}{64}=\frac{1}{4}\). Divide consecutive terms to find the ratio.
Step 3
Exam Tip
\(64,16,4,1,\ldots\) में \(\frac{16}{64}=\frac{1}{4}\) है। अनुपात के लिए लगातार पदों को भाग दें।
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गुणोत्तर श्रेढ़ी \(1,3,9,27,\ldots\) में (81) कौन-सा पद है?
In the geometric progression \(1,3,9,27,\ldots\), which term is (81)?
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A (4)वाँ / (4)th
B (5)वाँ / (5)th
C (6)वाँ / (6)th
D (7)वाँ / (7)th
Explanation opens after your attempt
Correct Answer
B. (5)वाँ / (5)th
Step 1
Concept
The terms are (1,3,9,27,81), so (81) is the fifth term. Identify powers in order.
Step 2
Why this answer is correct
The correct answer is B. (5)वाँ / (5)th. The terms are (1,3,9,27,81), so (81) is the fifth term. Identify powers in order.
Step 3
Exam Tip
पद (1,3,9,27,81) हैं, इसलिए (81) पाँचवाँ पद है। घातों को क्रम से पहचानें।
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यदि \(a_n=9\left(\frac{1}{3}\right)^{n-1}\) है, तो \(a_3\) क्या होगा?
If \(a_n=9\left(\frac{1}{3}\right)^{n-1}\), what is \(a_3\)?
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A (1)
B (2)
C (3)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(a_3=9\left(\frac{1}{3}\right)^2=1\). Calculate powers of fractional ratios carefully.
Step 2
Why this answer is correct
The correct answer is A. (1). \(a_3=9\left(\frac{1}{3}\right)^2=1\). Calculate powers of fractional ratios carefully.
Step 3
Exam Tip
\(a_3=9\left(\frac{1}{3}\right)^2=1\)। भिन्न अनुपात में घात की गणना सावधानी से करें।
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यदि (2,6,x,54) गुणोत्तर श्रेढ़ी है, तो (x) क्या होगा?
If (2,6,x,54) is a geometric progression, what is (x)?
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A (12)
B (18)
C (24)
D (30)
Explanation opens after your attempt
Step 1
Concept
The common ratio is \(\frac{6}{2}=3\), so \(x=6\cdot3=18\). Apply the same ratio to each next term.
Step 2
Why this answer is correct
The correct answer is B. (18). The common ratio is \(\frac{6}{2}=3\), so \(x=6\cdot3=18\). Apply the same ratio to each next term.
Step 3
Exam Tip
सार्व अनुपात \(\frac{6}{2}=3\) है, इसलिए \(x=6\cdot3=18\)। समान अनुपात को हर अगले पद पर लगाएं।
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किस अनुक्रम में सार्व अनुपात ऋणात्मक है?
Which sequence has a negative common ratio?
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A \(2,4,8,16,\ldots\)
B \(3,9,27,81,\ldots\)
C \(5,-10,20,-40,\ldots\)
D \(64,32,16,8,\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(5,-10,20,-40,\ldots\)
Step 1
Concept
In \(5,-10,20,-40,\ldots\), the common ratio is (-2). A negative ratio changes the signs of terms.
Step 2
Why this answer is correct
The correct answer is C. \(5,-10,20,-40,\ldots\). In \(5,-10,20,-40,\ldots\), the common ratio is (-2). A negative ratio changes the signs of terms.
Step 3
Exam Tip
\(5,-10,20,-40,\ldots\) में सार्व अनुपात (-2) है। ऋणात्मक अनुपात से पदों के चिह्न बदलते हैं।
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गुणोत्तर श्रेढ़ी \(10,20,40,80,\ldots\) के पहले (4) पदों का योग क्या है?
What is the sum of the first (4) terms of the geometric progression \(10,20,40,80,\ldots\)?
#sequences
#progressions
#geometric-progression
#sum
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A (140)
B (150)
C (160)
D (170)
Explanation opens after your attempt
Step 1
Concept
The sum is (10+20+40+80=150). For a small number of terms direct addition is easy.
Step 2
Why this answer is correct
The correct answer is B. (150). The sum is (10+20+40+80=150). For a small number of terms direct addition is easy.
Step 3
Exam Tip
योग (10+20+40+80=150) है। कम पद हों तो सीधे जोड़ना आसान है।
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यदि गुणोत्तर श्रेढ़ी में \(a_3=20\) और (r=2) है, तो \(a_1\) क्या होगा?
If a geometric progression has \(a_3=20\) and (r=2), what is \(a_1\)?
#sequences
#progressions
#geometric-progression
#first-term
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A (4)
B (5)
C (10)
D (40)
Explanation opens after your attempt
Step 1
Concept
\(a_3=a_1r^2\), so \(20=a_1\cdot4\) and \(a_1=5\). Divide by \(r^2\) to move two terms backward.
Step 2
Why this answer is correct
The correct answer is B. (5). \(a_3=a_1r^2\), so \(20=a_1\cdot4\) and \(a_1=5\). Divide by \(r^2\) to move two terms backward.
Step 3
Exam Tip
\(a_3=a_1r^2\), इसलिए \(20=a_1\cdot4\) और \(a_1=5\)। दो पद पीछे जाने के लिए \(r^2\) से भाग दें।
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अनुक्रम \(6,18,54,162,\ldots\) का (n)वाँ पद कौन-सा है?
What is the (n)th term of the sequence \(6,18,54,162,\ldots\)?
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#general-term
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A \(a_n=6\cdot3^{n-1}\)
B \(a_n=3\cdot6^{n-1}\)
C \(a_n=6n+3\)
D \(a_n=18\cdot3^{n-1}\)
Explanation opens after your attempt
Correct Answer
A. \(a_n=6\cdot3^{n-1}\)
Step 1
Concept
The first term is (6) and the ratio is (3), so \(a_n=6\cdot3^{n-1}\). Use both first term and ratio in the general term.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=6\cdot3^{n-1}\). The first term is (6) and the ratio is (3), so \(a_n=6\cdot3^{n-1}\). Use both first term and ratio in the general term.
Step 3
Exam Tip
पहला पद (6) और अनुपात (3) है, इसलिए \(a_n=6\cdot3^{n-1}\)। सामान्य पद में पहला पद और अनुपात दोनों रखें।
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यदि (a=1) और (r=2) है, तो पहले (6) पदों का योग क्या होगा?
If (a=1) and (r=2), what will be the sum of the first (6) terms?
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#sum
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A (31)
B (63)
C (64)
D (127)
Explanation opens after your attempt
Step 1
Concept
The first (6) terms are (1,2,4,8,16,32), and their sum is (63). Add powers of (2) carefully.
Step 2
Why this answer is correct
The correct answer is B. (63). The first (6) terms are (1,2,4,8,16,32), and their sum is (63). Add powers of (2) carefully.
Step 3
Exam Tip
पहले (6) पद (1,2,4,8,16,32) हैं और योग (63) है। (2) की घातों के योग को ध्यान से जोड़ें।
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यदि \(a_1=12\) और \(r=\frac{1}{2}\) है, तो \(a_4\) क्या होगा?
If \(a_1=12\) and \(r=\frac{1}{2}\), what is \(a_4\)?
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#progressions
#geometric-progression
#fraction-ratio
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A \(\frac{3}{2}\)
B (2)
C (3)
D (6)
Explanation opens after your attempt
Correct Answer
A. \(\frac{3}{2}\)
Step 1
Concept
(a_4=12\left\(\frac{1}{2}\right\)3 =\frac{3}{2}). With a fractional ratio each term becomes half.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{3}{2}\). (a_4=12\left\(\frac{1}{2}\right\)3 =\frac{3}{2}). With a fractional ratio each term becomes half.
Step 3
Exam Tip
(a_4=12\left\(\frac{1}{2}\right\)3 =\frac{3}{2})। भिन्न अनुपात में हर बार आधा होता है।
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कौन-सा (x) मान (4,20,x) को गुणोत्तर श्रेढ़ी बनाता है?
Which value of (x) makes (4,20,x) a geometric progression?
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#missing-term
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A (80)
B (90)
C (100)
D (120)
Explanation opens after your attempt
Step 1
Concept
The common ratio is \(\frac{20}{4}=5\), so \(x=20\cdot5=100\). Multiply the second term by the ratio to get the third term.
Step 2
Why this answer is correct
The correct answer is C. (100). The common ratio is \(\frac{20}{4}=5\), so \(x=20\cdot5=100\). Multiply the second term by the ratio to get the third term.
Step 3
Exam Tip
सार्व अनुपात \(\frac{20}{4}=5\) है, इसलिए \(x=20\cdot5=100\)। तीसरे पद के लिए दूसरे पद को अनुपात से गुणा करें।
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अनुक्रम \(2,8,32,128,\ldots\) में (512) कौन-सा पद है?
In the sequence \(2,8,32,128,\ldots\), which term is (512)?
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#progressions
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#term-position
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A (4)वाँ / (4)th
B (5)वाँ / (5)th
C (6)वाँ / (6)th
D (7)वाँ / (7)th
Explanation opens after your attempt
Correct Answer
B. (5)वाँ / (5)th
Step 1
Concept
The common ratio is (4), and the terms are (2,8,32,128,512). For position form terms in order.
Step 2
Why this answer is correct
The correct answer is B. (5)वाँ / (5)th. The common ratio is (4), and the terms are (2,8,32,128,512). For position form terms in order.
Step 3
Exam Tip
सार्व अनुपात (4) है और पद (2,8,32,128,512) हैं। पद-संख्या के लिए क्रम से पद बनाएं।
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यदि किसी गुणोत्तर श्रेढ़ी में \(a_2=18\) और \(a_4=162\) है, तो (r) क्या होगा?
If a geometric progression has \(a_2=18\) and \(a_4=162\), what is (r)?
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#progressions
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A (2)
B (3)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(a_4=a_2r^2\), so \(162=18r^2\) and \(r^2=9\), giving positive ratio (r=3). Use the gap between terms as the exponent.
Step 2
Why this answer is correct
The correct answer is B. (3). \(a_4=a_2r^2\), so \(162=18r^2\) and \(r^2=9\), giving positive ratio (r=3). Use the gap between terms as the exponent.
Step 3
Exam Tip
\(a_4=a_2r^2\), इसलिए \(162=18r^2\) और \(r^2=9\), धनात्मक अनुपात (r=3) है। पदों के बीच दूरी को घात बनाएं।
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गुणोत्तर श्रेढ़ी \(1,-2,4,-8,\ldots\) का पाँचवाँ पद क्या है?
What is the fifth term of the geometric progression \(1,-2,4,-8,\ldots\)?
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#progressions
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#negative-ratio
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A (8)
B (16)
C (-16)
D (-32)
Explanation opens after your attempt
Step 1
Concept
The common ratio is (-2), so the fifth term is ((-8)(-2)=16). With a negative ratio signs alternate.
Step 2
Why this answer is correct
The correct answer is B. (16). The common ratio is (-2), so the fifth term is ((-8)(-2)=16). With a negative ratio signs alternate.
Step 3
Exam Tip
सार्व अनुपात (-2) है, इसलिए पाँचवाँ पद ((-8)(-2)=16) है। ऋणात्मक अनुपात में चिह्न बदलता है।
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किस विकल्प में (a=9) और (r=3) वाली गुणोत्तर श्रेढ़ी के पहले चार पद हैं?
Which option has the first four terms of a geometric progression with (a=9) and (r=3)?
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A (9,12,15,18)
B (9,18,36,72)
C (9,27,81,243)
D (3,9,27,81)
Explanation opens after your attempt
Correct Answer
C. (9,27,81,243)
Step 1
Concept
The first term is (9), and multiplying by (3) each time gives (9,27,81,243). Match both (a) and (r).
Step 2
Why this answer is correct
The correct answer is C. (9,27,81,243). The first term is (9), and multiplying by (3) each time gives (9,27,81,243). Match both (a) and (r).
Step 3
Exam Tip
पहला पद (9) है और हर बार (3) से गुणा करने पर (9,27,81,243) मिलते हैं। (a) और (r) दोनों को मिलाएं।
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यदि \(5,15,45,\ldots\) गुणोत्तर श्रेढ़ी है, तो पहले तीन पदों का गुणनफल क्या है?
If \(5,15,45,\ldots\) is a geometric progression, what is the product of the first three terms?
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#product
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A (3375)
B (3000)
C (2500)
D (2025)
Explanation opens after your attempt
Step 1
Concept
The product is \(5\cdot15\cdot45=3375\). Multiply each term carefully in product questions.
Step 2
Why this answer is correct
The correct answer is A. (3375). The product is \(5\cdot15\cdot45=3375\). Multiply each term carefully in product questions.
Step 3
Exam Tip
गुणनफल \(5\cdot15\cdot45=3375\) है। गुणनफल में हर पद को सावधानी से गुणा करें।
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गुणोत्तर श्रेढ़ी \(100,50,25,\ldots\) में अगला पद क्या होगा?
What is the next term in the geometric progression \(100,50,25,\ldots\)?
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#next-term
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A (10)
B \(\frac{25}{2}\)
C (15)
D (20)
Explanation opens after your attempt
Correct Answer
B. \(\frac{25}{2}\)
Step 1
Concept
The common ratio is \(\frac{1}{2}\), so the next term is \(25\cdot\frac{1}{2}=\frac{25}{2}\). Multiply by the ratio.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{25}{2}\). The common ratio is \(\frac{1}{2}\), so the next term is \(25\cdot\frac{1}{2}=\frac{25}{2}\). Multiply by the ratio.
Step 3
Exam Tip
सार्व अनुपात \(\frac{1}{2}\) है, इसलिए अगला पद \(25\cdot\frac{1}{2}=\frac{25}{2}\) होगा। अनुपात से गुणा करें।
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यदि \(a_n=4\cdot2^{n-1}\) है, तो यह कौन-सा अनुक्रम बनाता है?
If \(a_n=4\cdot2^{n-1}\), which sequence does it form?
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#sequence-from-rule
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A \(4,6,8,10,\ldots\)
B \(2,4,8,16,\ldots\)
C \(4,8,16,32,\ldots\)
D \(8,16,32,64,\ldots\)
Explanation opens after your attempt
Correct Answer
C. \(4,8,16,32,\ldots\)
Step 1
Concept
Putting (n=1,2,3,4) gives (4,8,16,32). Form the initial terms from the rule and match the option.
Step 2
Why this answer is correct
The correct answer is C. \(4,8,16,32,\ldots\). Putting (n=1,2,3,4) gives (4,8,16,32). Form the initial terms from the rule and match the option.
Step 3
Exam Tip
(n=1,2,3,4) रखने पर (4,8,16,32) मिलते हैं। नियम से शुरू के पद बनाकर विकल्प मिलाएं।
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यदि (x,12,36) गुणोत्तर श्रेढ़ी के लगातार पद हैं, तो (x) क्या होगा?
If (x,12,36) are consecutive terms of a geometric progression, what is (x)?
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(\frac{36}{12}=3\), so \(\frac{12}{x}=3\) and (x=4). Keep consecutive ratios equal.
Step 2
Why this answer is correct
The correct answer is C. (4). \(\frac{36}{12}=3\), so \(\frac{12}{x}=3\) and (x=4). Keep consecutive ratios equal.
Step 3
Exam Tip
\(\frac{36}{12}=3\), इसलिए \(\frac{12}{x}=3\) और (x=4)। लगातार अनुपात बराबर रखें।
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गुणोत्तर श्रेढ़ी \(8,24,72,\ldots\) का (6)वाँ पद क्या है?
What is the (6)th term of the geometric progression \(8,24,72,\ldots\)?
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#nth-term
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A (648)
B (972)
C (1944)
D (5832)
Explanation opens after your attempt
Step 1
Concept
Here (r=3), so \(a_6=8\cdot3^5=1944\). The sixth term uses the ratio (5) times.
Step 2
Why this answer is correct
The correct answer is C. (1944). Here (r=3), so \(a_6=8\cdot3^5=1944\). The sixth term uses the ratio (5) times.
Step 3
Exam Tip
यहाँ (r=3), इसलिए \(a_6=8\cdot3^5=1944\)। (6)वें पद में (5) बार अनुपात लगता है।
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किस अनुक्रम का सार्व अनुपात (5) है?
Which sequence has common ratio (5)?
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A \(5,10,15,20,\ldots\)
B \(2,10,50,250,\ldots\)
C \(1,5,25,100,\ldots\)
D \(25,5,1,\frac{1}{5},\ldots\)
Explanation opens after your attempt
Correct Answer
B. \(2,10,50,250,\ldots\)
Step 1
Concept
In \(2,10,50,250,\ldots\), each term is multiplied by (5). For the ratio divide the next term by the previous term.
Step 2
Why this answer is correct
The correct answer is B. \(2,10,50,250,\ldots\). In \(2,10,50,250,\ldots\), each term is multiplied by (5). For the ratio divide the next term by the previous term.
Step 3
Exam Tip
\(2,10,50,250,\ldots\) में हर बार (5) से गुणा हो रहा है। अनुपात के लिए अगले पद को पिछले पद से भाग दें।
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यदि \(a_1= \frac{3}{4}\) और (r=2) है, तो \(a_4\) क्या होगा?
If \(a_1= \frac{3}{4}\) and (r=2), what is \(a_4\)?
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A (3)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(a_4=\frac{3}{4}\cdot2^3=6\). Multiply the power carefully with the fractional first term.
Step 2
Why this answer is correct
The correct answer is C. (6). \(a_4=\frac{3}{4}\cdot2^3=6\). Multiply the power carefully with the fractional first term.
Step 3
Exam Tip
\(a_4=\frac{3}{4}\cdot2^3=6\)। भिन्न पहले पद के साथ घात को सावधानी से गुणा करें।
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गुणोत्तर श्रेढ़ी \(2,4,8,16,\ldots\) के पहले (5) पदों का योग क्या है?
What is the sum of the first (5) terms of the geometric progression \(2,4,8,16,\ldots\)?
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#progressions
#geometric-progression
#sum
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A (30)
B (32)
C (62)
D (64)
Explanation opens after your attempt
Step 1
Concept
The first (5) terms are (2,4,8,16,32), and the sum is (62). Write the terms in order and add.
Step 2
Why this answer is correct
The correct answer is C. (62). The first (5) terms are (2,4,8,16,32), and the sum is (62). Write the terms in order and add.
Step 3
Exam Tip
पहले (5) पद (2,4,8,16,32) हैं और योग (62) है। पदों को क्रम से लिखकर जोड़ें।
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यदि \(a_1=3\) और (r=-2) है, तो पहले चार पद कौन-से हैं?
If \(a_1=3\) and (r=-2), what are the first four terms?
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#negative-ratio
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A (3,6,12,24)
B (3,-6,12,-24)
C (3,-2,4,-8)
D (-3,6,-12,24)
Explanation opens after your attempt
Correct Answer
B. (3,-6,12,-24)
Step 1
Concept
Multiplying by (-2) each time gives (3,-6,12,-24). A negative ratio makes the signs alternate.
Step 2
Why this answer is correct
The correct answer is B. (3,-6,12,-24). Multiplying by (-2) each time gives (3,-6,12,-24). A negative ratio makes the signs alternate.
Step 3
Exam Tip
हर बार (-2) से गुणा करने पर (3,-6,12,-24) मिलते हैं। ऋणात्मक अनुपात से चिह्न बदलते हैं।
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किस (x) के लिए (8,x,18) धनात्मक गुणोत्तर श्रेढ़ी के लगातार पद होंगे?
For which (x) will (8,x,18) be consecutive terms of a positive geometric progression?
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#geometric-mean
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A (10)
B (12)
C (14)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(x^2=8\cdot18=144\), so positive (x=12). The middle term is the geometric mean.
Step 2
Why this answer is correct
The correct answer is B. (12). \(x^2=8\cdot18=144\), so positive (x=12). The middle term is the geometric mean.
Step 3
Exam Tip
\(x^2=8\cdot18=144\), इसलिए धनात्मक (x=12)। मध्य पद ज्यामितीय माध्य होता है।
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अनुक्रम \(7,21,63,189,\ldots\) का (n)वाँ पद कौन-सा है?
What is the (n)th term of the sequence \(7,21,63,189,\ldots\)?
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#general-term
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A \(a_n=7\cdot3^{n-1}\)
B \(a_n=3\cdot7^{n-1}\)
C \(a_n=7n+3\)
D \(a_n=21\cdot3^{n-1}\)
Explanation opens after your attempt
Correct Answer
A. \(a_n=7\cdot3^{n-1}\)
Step 1
Concept
The first term is (7) and the ratio is (3), so \(a_n=7\cdot3^{n-1}\). Use the first term and ratio in the general term.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=7\cdot3^{n-1}\). The first term is (7) and the ratio is (3), so \(a_n=7\cdot3^{n-1}\). Use the first term and ratio in the general term.
Step 3
Exam Tip
पहला पद (7) और अनुपात (3) है, इसलिए \(a_n=7\cdot3^{n-1}\)। सामान्य पद में पहला पद और अनुपात रखें।
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यदि \(a_2=24\) और \(a_5=192\) है, तथा अनुपात धनात्मक है, तो (r) क्या है?
If \(a_2=24\) and \(a_5=192\), and the ratio is positive, what is (r)?
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#geometric-progression
#common-ratio
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(a_5=a_2r^3\), so \(192=24r^3\) and \(r^3=8\), hence (r=2). The gap between positions decides the exponent.
Step 2
Why this answer is correct
The correct answer is A. (2). \(a_5=a_2r^3\), so \(192=24r^3\) and \(r^3=8\), hence (r=2). The gap between positions decides the exponent.
Step 3
Exam Tip
\(a_5=a_2r^3\), इसलिए \(192=24r^3\) और \(r^3=8\), अतः (r=2)। पदों की दूरी घात तय करती है।
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गुणोत्तर श्रेढ़ी \(3,12,48,\ldots\) में \(a_5\) क्या होगा?
In the geometric progression \(3,12,48,\ldots\), what is \(a_5\)?
#sequences
#progressions
#geometric-progression
#nth-term
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A (192)
B (384)
C (768)
D (1024)
Explanation opens after your attempt
Step 1
Concept
The common ratio is (4), so \(a_5=3\cdot4^4=768\). For the fifth term the ratio is used (4) times.
Step 2
Why this answer is correct
The correct answer is C. (768). The common ratio is (4), so \(a_5=3\cdot4^4=768\). For the fifth term the ratio is used (4) times.
Step 3
Exam Tip
सार्व अनुपात (4) है, इसलिए \(a_5=3\cdot4^4=768\)। (5)वें पद के लिए (4) बार अनुपात लगता है।
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यदि गुणोत्तर श्रेढ़ी में \(a_1=2\) और \(a_3=50\) है, तो धनात्मक (r) क्या होगा?
If a geometric progression has \(a_1=2\) and \(a_3=50\), what is the positive (r)?
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#progressions
#geometric-progression
#common-ratio
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(a_3=a_1r^2\), so \(50=2r^2\) and \(r^2=25\), hence positive (r=5). When positive ratio is asked take the positive root.
Step 2
Why this answer is correct
The correct answer is C. (5). \(a_3=a_1r^2\), so \(50=2r^2\) and \(r^2=25\), hence positive (r=5). When positive ratio is asked take the positive root.
Step 3
Exam Tip
\(a_3=a_1r^2\), इसलिए \(50=2r^2\) और \(r^2=25\), अतः धनात्मक (r=5)। धनात्मक अनुपात मांगने पर धनात्मक मूल लें।
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कौन-सा कथन \(12,6,3,\frac{3}{2},\ldots\) के लिए सही है?
Which statement is correct for \(12,6,3,\frac{3}{2},\ldots\)?
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A यह (r=2) वाली गुणोत्तर श्रेढ़ी है / It is a geometric progression with (r=2)
B यह \(r=\frac{1}{2}\) वाली गुणोत्तर श्रेढ़ी है / It is a geometric progression with \(r=\frac{1}{2}\)
C यह समान अंतर वाली श्रेढ़ी है / It has a constant difference
D यह गुणोत्तर श्रेढ़ी नहीं है / It is not a geometric progression
Explanation opens after your attempt
Correct Answer
B. यह \(r=\frac{1}{2}\) वाली गुणोत्तर श्रेढ़ी है / It is a geometric progression with \(r=\frac{1}{2}\)
Step 1
Concept
Each term is half of the previous term, so \(r=\frac{1}{2}\). To check the ratio divide the next term by the previous term.
Step 2
Why this answer is correct
The correct answer is B. यह \(r=\frac{1}{2}\) वाली गुणोत्तर श्रेढ़ी है / It is a geometric progression with \(r=\frac{1}{2}\). Each term is half of the previous term, so \(r=\frac{1}{2}\). To check the ratio divide the next term by the previous term.
Step 3
Exam Tip
हर पद पिछले पद का आधा है, इसलिए \(r=\frac{1}{2}\) है। अनुपात जांचने के लिए अगला पद पिछले पद से भाग दें।
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यदि \(1,4,16,64,\ldots\) में \(a_n=256\) है, तो (n) क्या है?
If \(a_n=256\) in \(1,4,16,64,\ldots\), what is (n)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The terms are (1,4,16,64,256), so (n=5). For position identify powers in order.
Step 2
Why this answer is correct
The correct answer is B. (5). The terms are (1,4,16,64,256), so (n=5). For position identify powers in order.
Step 3
Exam Tip
पद (1,4,16,64,256) हैं, इसलिए (n=5)। पद-संख्या के लिए क्रम से घात पहचानें।
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गुणोत्तर श्रेढ़ी \(2,6,18,54,\ldots\) में \(a_3+a_4\) कितना है?
In the geometric progression \(2,6,18,54,\ldots\), what is \(a_3+a_4\)?
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#sum-of-terms
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A (60)
B (66)
C (72)
D (78)
Explanation opens after your attempt
Step 1
Concept
\(a_3=18\) and \(a_4=54\), so the sum is (72). Find both terms separately and then add.
Step 2
Why this answer is correct
The correct answer is C. (72). \(a_3=18\) and \(a_4=54\), so the sum is (72). Find both terms separately and then add.
Step 3
Exam Tip
\(a_3=18\) और \(a_4=54\), इसलिए योग (72) है। दोनों पद अलग निकालकर जोड़ें।
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यदि गुणोत्तर श्रेढ़ी में (a=16) और \(r=\frac{1}{2}\) है, तो पहले चार पदों का योग क्या होगा?
If a geometric progression has (a=16) and \(r=\frac{1}{2}\), what is the sum of the first four terms?
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#sum
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A (24)
B (28)
C (30)
D (32)
Explanation opens after your attempt
Step 1
Concept
The first four terms are (16,8,4,2), and the sum is (30). In a decreasing geometric progression write the terms in order.
Step 2
Why this answer is correct
The correct answer is C. (30). The first four terms are (16,8,4,2), and the sum is (30). In a decreasing geometric progression write the terms in order.
Step 3
Exam Tip
पहले चार पद (16,8,4,2) हैं और योग (30) है। घटती गुणोत्तर श्रेढ़ी में पद क्रम से लिखें।
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