progressions se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
1793 questions tagged with progressions.
D. (45)
पहले चार पद (24,12,6,3) हैं और योग (45) है। घटती गुणोत्तर श्रेढ़ी में पद क्रम से लिखें। / The first four terms are (24,12,6,3), and the sum is (45). In a decreasing geometric progression write the terms in order.
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Which statement is correct for \(18,6,2,\frac{2}{3},\ldots\)?
B. यह \(r=\frac{1}{3}\) वाली गुणोत्तर श्रेढ़ी हैIt is a geometric progression with \(r=\frac{1}{3}\)
हर पद पिछले पद का \(\frac{1}{3}\) है, इसलिए \(r=\frac{1}{3}\) है। अनुपात जांचने के लिए अगला पद पिछले पद से भाग दें। / Each term is \(\frac{1}{3}\) of the previous term, so \(r=\frac{1}{3}\). To check the ratio divide the next term by the previous term.
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If a geometric progression has \(a_1=3\) and \(a_3=75\), what is the positive (r)?
C. (5)
\(a_3=a_1r^2\), इसलिए \(75=3r^2\) और \(r^2=25\), अतः धनात्मक (r=5)। धनात्मक अनुपात मांगने पर धनात्मक मूल लें। / \(a_3=a_1r^2\), so \(75=3r^2\) and \(r^2=25\), hence positive (r=5). When positive ratio is asked take the positive root.
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In the geometric progression \(4,16,64,\ldots\), what is \(a_5\)?
C. (1024)
सार्व अनुपात (4) है, इसलिए \(a_5=4\cdot4^4=1024\)। (5)वें पद के लिए (4) बार अनुपात लगता है। / The common ratio is (4), so \(a_5=4\cdot4^4=1024\). For the fifth term the ratio is used (4) times.
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If \(a_2=30\) and \(a_5=810\), and the ratio is positive, what is (r)?
B. (3)
\(a_5=a_2r^3\), इसलिए \(810=30r^3\) और \(r^3=27\), अतः (r=3)। पदों की दूरी घात तय करती है। / \(a_5=a_2r^3\), so \(810=30r^3\) and \(r^3=27\), hence (r=3). The gap between positions decides the exponent.
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What is the (n)th term of the sequence \(11,33,99,297,\ldots\)?
A. \(a_n=11\cdot3^{n-1}\)
पहला पद (11) और अनुपात (3) है, इसलिए \(a_n=11\cdot3^{n-1}\)। सामान्य पद में पहला पद और अनुपात रखें। / The first term is (11) and the ratio is (3), so \(a_n=11\cdot3^{n-1}\). Use the first term and ratio in the general term.
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If \(a_1=4\) and (r=-3), what are the first four terms?
B. (4,-12,36,-108)
हर बार (-3) से गुणा करने पर (4,-12,36,-108) मिलते हैं। ऋणात्मक अनुपात से चिह्न बदलते हैं। / Multiplying by (-3) each time gives (4,-12,36,-108). A negative ratio makes the signs alternate.
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Which sequence has common ratio (4)?
B. \(3,12,48,192,\ldots\)
\(3,12,48,192,\ldots\) में हर बार (4) से गुणा हो रहा है। अनुपात के लिए अगले पद को पिछले पद से भाग दें। / In \(3,12,48,192,\ldots\), each term is multiplied by (4). For the ratio divide the next term by the previous term.
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If \(a_n=6\cdot3^{n-1}\), which sequence does it form?
C. \(6,18,54,162,\ldots\)
(n=1,2,3,4) रखने पर (6,18,54,162) मिलते हैं। नियम से शुरू के पद बनाकर विकल्प मिलाएं। / Putting (n=1,2,3,4) gives (6,18,54,162). Form the initial terms from the rule and match the option.
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What is the next term in the geometric progression \(72,24,8,\ldots\)?
B. \(\frac{8}{3}\)
सार्व अनुपात \(\frac{1}{3}\) है, इसलिए अगला पद \(8\cdot\frac{1}{3}=\frac{8}{3}\) होगा। अनुपात से गुणा करें। / The common ratio is \(\frac{1}{3}\), so the next term is \(8\cdot\frac{1}{3}=\frac{8}{3}\). Multiply by the ratio.
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Which option has the first four terms of a geometric progression with (a=12) and (r=2)?
B. (12,24,48,96)
पहला पद (12) है और हर बार (2) से गुणा करने पर (12,24,48,96) मिलते हैं। (a) और (r) दोनों को मिलाएं। / The first term is (12), and multiplying by (2) each time gives (12,24,48,96). Match both (a) and (r).
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What is the sixth term of the geometric progression \(2,-4,8,-16,\ldots\)?
C. (-64)
सार्व अनुपात (-2) है और पद (2,-4,8,-16,32,-64) हैं। ऋणात्मक अनुपात में चिह्न बदलता है। / The common ratio is (-2), and the terms are (2,-4,8,-16,32,-64). With a negative ratio signs alternate.
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If a geometric progression has \(a_2=12\) and \(a_5=324\), and the ratio is positive, what is (r)?
B. (3)
\(a_5=a_2r^3\), इसलिए \(324=12r^3\) और \(r^3=27\), अतः (r=3)। पदों की दूरी घात तय करती है। / \(a_5=a_2r^3\), so \(324=12r^3\) and \(r^3=27\), hence (r=3). The gap between positions decides the exponent.
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In the sequence \(3,15,75,375,\ldots\), which term is (1875)?
B. (5)वाँ(5)th
सार्व अनुपात (5) है और पद (3,15,75,375,1875) हैं। पद-संख्या के लिए क्रम से पद बनाएं। / The common ratio is (5), and the terms are (3,15,75,375,1875). Form terms in order to find the position.
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Which value of (x) makes (6,30,x) a geometric progression?
C. (150)
सार्व अनुपात \(\frac{30}{6}=5\) है, इसलिए \(x=30\cdot5=150\)। तीसरे पद के लिए दूसरे पद को अनुपात से गुणा करें। / The common ratio is \(\frac{30}{6}=5\), so \(x=30\cdot5=150\). Multiply the second term by the ratio to get the third term.
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If \(a_1=20\) and \(r=\frac{1}{2}\), what is \(a_4\)?
B. \(\frac{5}{2}\)
\(a_4=20\left(\frac{1}{2}\right)^3=\frac{5}{2}\)। भिन्न अनुपात में हर बार आधा होता है। / \(a_4=20\left(\frac{1}{2}\right)^3=\frac{5}{2}\). With a fractional ratio each term becomes half.
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What is the (n)th term of the sequence \(10,20,40,80,\ldots\)?
A. \(a_n=10\cdot2^{n-1}\)
पहला पद (10) और अनुपात (2) है, इसलिए \(a_n=10\cdot2^{n-1}\)। सामान्य पद में पहला पद और अनुपात दोनों रखें। / The first term is (10) and the ratio is (2), so \(a_n=10\cdot2^{n-1}\). Use both first term and ratio in the general term.
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If (3,12,x,192) is a geometric progression, what is (x)?
C. (48)
सार्व अनुपात \(\frac{12}{3}=4\) है, इसलिए \(x=12\cdot4=48\)। समान अनुपात को हर अगले पद पर लगाएं। / The common ratio is \(\frac{12}{3}=4\), so \(x=12\cdot4=48\). Apply the same ratio to each next term.
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In the geometric progression \(2,6,18,54,\ldots\), which term is (486)?
C. (6)वाँ(6)th
पद (2,6,18,54,162,486) हैं, इसलिए (486) छठा पद है। पद-संख्या के लिए क्रम से पद बनाएं। / The terms are (2,6,18,54,162,486), so (486) is the sixth term. Form terms in order to find the position.
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Which option has common ratio \(\frac{1}{5}\)?
C. \(100,20,4,\frac{4}{5},\ldots\)
\(100,20,4,\frac{4}{5},\ldots\) में \(\frac{20}{100}=\frac{1}{5}\) है। अनुपात के लिए लगातार पदों को भाग दें। / In \(100,20,4,\frac{4}{5},\ldots\), \(\frac{20}{100}=\frac{1}{5}\). Divide consecutive terms to find the ratio.
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What is the sixth term of the geometric progression \(128,64,32,16,\ldots\)?
A. (4)
सार्व अनुपात \(\frac{1}{2}\) है, इसलिए पद (128,64,32,16,8,4) होंगे। घटती श्रेढ़ी में बार-बार अनुपात से गुणा करें। / The common ratio is \(\frac{1}{2}\), so the terms are (128,64,32,16,8,4). In a decreasing progression multiply by the ratio repeatedly.
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What is the (6)th term of the sequence \(\frac{3}{4},\frac{3}{2},3,6,\ldots\)?
B. (24)
सार्व अनुपात (2) है और पद \(\frac{3}{4},\frac{3}{2},3,6,12,24\) हैं। क्रम से गुणा करते जाएं। / The common ratio is (2), and the terms are \(\frac{3}{4},\frac{3}{2},3,6,12,24\). Keep multiplying in order.
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If a geometric progression has \(a_1=2\) and \(a_5=162\), and the ratio is positive, what is (r)?
B. (3)
\(a_5=a_1r^4\), इसलिए \(162=2r^4\) और \(r^4=81\), अतः (r=3)। पद-संख्या का अंतर घात तय करता है। / \(a_5=a_1r^4\), so \(162=2r^4\) and \(r^4=81\), hence (r=3). The position gap decides the exponent.
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What is the next term in the geometric progression \(2,10,50,\ldots\)?
B. (250)
सार्व अनुपात (5) है, इसलिए अगला पद \(50\cdot5=250\) होगा। अगले पद के लिए पिछले पद को अनुपात से गुणा करें। / The common ratio is (5), so the next term is \(50\cdot5=250\). Multiply the previous term by the ratio to get the next term.
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If a geometric progression has (a=8) and (r=3), what are the first four terms?
D. (8,24,72,216)
पहला पद (8) है और हर बार (3) से गुणा करने पर (8,24,72,216) मिलते हैं। गुणोत्तर श्रेढ़ी में अगला पद गुणा करके बनता है। / The first term is (8), and multiplying by (3) each time gives (8,24,72,216). In a geometric progression each next term is formed by multiplication.
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In the geometric progression \(9,27,81,243,\ldots\), which term is (729)?
B. (5)वाँ(5)th
पद (9,27,81,243,729) हैं, इसलिए (729) पाँचवाँ पद है। पद-संख्या के लिए पदों को क्रम से बनाएं। / The terms are (9,27,81,243,729), so (729) is the fifth term. Form the terms in order to find the position.
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If \(a_n=3\cdot2^{n-1}\), what is the value of \(a_6\)?
C. 96
दिया है \(a_n=3\cdot2^{n-1}\)। \(n=6\) रखने पर \(a_6=3\cdot2^{6-1}=3\cdot2^5=3\cdot32=96\)। इसलिए सही उत्तर 96 है। 64 केवल \(2^6\) है, इसमें प्रारम्भिक गुणक 3 शामिल नहीं है। परीक्षा टिप: पद ज्ञात करते समय पहले \(n\) का मान सूत्र में रखकर घात \(n-1\) को सावधानी से सरल करें। / Given \(a_n=3\cdot2^{n-1}\), substitute \(n=6\): \(a_6=3\cdot2^{6-1}=3\cdot2^5=3\cdot32=96\). Hence, 96 is correct. The value 64 is only \(2^6\); it ignores the initial factor 3. Exam tip: Substitute the term number first, then simplify the exponent \(n-1\) carefully.
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What is the common ratio of the sequence \(96,48,24,12,\ldots\)?
D. \(\frac{1}{2}\)
सार्व अनुपात \(\frac{48}{96}=\frac{1}{2}\) है। घटती गुणोत्तर श्रेढ़ी में अनुपात (1) से छोटा हो सकता है। / The common ratio is \(\frac{48}{96}=\frac{1}{2}\). In a decreasing geometric progression the ratio can be less than (1).
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What is the common ratio in the sequence \(5,15,45,135,\ldots\)?
C. (3)
सार्व अनुपात \(\frac{15}{5}=3\) है। गुणोत्तर श्रेढ़ी में अगला पद पिछले पद से भाग दें। / The common ratio is \(\frac{15}{5}=3\). In a geometric progression divide the next term by the previous term.
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Which geometric progression has first term (6) and common ratio (4)?
A. \(6,24,96,384,\ldots\)
पहला पद (6) है और हर बार (4) से गुणा हो रहा है। परीक्षा में पहला पद और अनुपात दोनों मिलाएँ। / The first term is (6) and each term is multiplied by (4). In exams match both the first term and the ratio.
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