Level 54 • 50/50 questions • 35 seconds per question.
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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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If a geometric progression has first term (4) and common ratio (5), what is the third term?
A. (100)
\(a_3=4\cdot5^2=100\)। (n)वें पद में अनुपात की घात (n-1) होती है। / \(a_3=4\cdot5^2=100\). In the (n)th term the exponent of the ratio is (n-1).
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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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If \(a_n=3\cdot2^{n-1}\), what is the value of \(a_6\)?
C. (96)
\(a_6=3\cdot2^5=96\)। घात में (n-1) का मान पहले निकालें। / \(a_6=3\cdot2^5=96\). First find the value of (n-1) in the exponent.
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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 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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What is the common ratio of the sequence \(4,-12,36,-108,\ldots\)?
A. (-3)
सार्व अनुपात \(\frac{-12}{4}=-3\) है। ऋणात्मक अनुपात से पदों के चिह्न बदलते हैं। / The common ratio is \(\frac{-12}{4}=-3\). A negative ratio changes the signs of terms.
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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 (4,x,64) are consecutive positive terms of a geometric progression, what is (x)?
A. (16)
\(x^2=4\cdot64=256\), इसलिए धनात्मक (x=16)। मध्य पद ज्यामितीय माध्य होता है। / \(x^2=4\cdot64=256\), so positive (x=16). The middle term is the geometric mean.
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What is the sum of the first (4) terms of the geometric progression \(6,18,54,162,\ldots\)?
C. (240)
योग (6+18+54+162=240) है। कम पद हों तो सीधे जोड़ना सुरक्षित है। / The sum is (6+18+54+162=240). For a small number of terms direct addition is safe.
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If a geometric progression has \(a_2=21\) and (r=3), what is \(a_1\)?
D. (7)
\(a_2=a_1r\), इसलिए \(21=a_1\cdot3\) और \(a_1=7\)। पीछे जाने के लिए अनुपात से भाग दें। / \(a_2=a_1r\), so \(21=a_1\cdot3\) and \(a_1=7\). Divide by the ratio to move backward.
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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 (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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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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If (a=2) and (r=6), what is \(a_4\)?
C. (432)
\(a_4=2\cdot6^3=432\)। (n)वें पद में घात (n-1) होती है। / \(a_4=2\cdot6^3=432\). In the (n)th term the exponent is (n-1).
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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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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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If \(a_n=16\left(\frac{1}{2}\right)^{n-1}\), what is \(a_4\)?
B. (2)
\(a_4=16\left(\frac{1}{2}\right)^3=2\)। भिन्न अनुपात में घात की गणना सावधानी से करें। / \(a_4=16\left(\frac{1}{2}\right)^3=2\). Calculate powers of fractional ratios carefully.
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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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What is the sum of the first (4) terms of the geometric progression \(12,24,48,96,\ldots\)?
A. (180)
योग (12+24+48+96=180) है। कम पद हों तो सीधे जोड़ना आसान है। / The sum is (12+24+48+96=180). For a small number of terms direct addition is easy.
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If a geometric progression has \(a_3=45\) and (r=3), what is \(a_1\)?
B. (5)
\(a_3=a_1r^2\), इसलिए \(45=a_1\cdot9\) और \(a_1=5\)। दो पद पीछे जाने के लिए \(r^2\) से भाग दें। / \(a_3=a_1r^2\), so \(45=a_1\cdot9\) and \(a_1=5\). Divide by \(r^2\) to move two terms backward.
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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 (a=2) and (r=3), what will be the sum of the first (5) terms?
B. (242)
पहले (5) पद (2,6,18,54,162) हैं और योग (242) है। पदों को क्रम से लिखकर जोड़ें। / The first (5) terms are (2,6,18,54,162), and their sum is (242). Write the terms in order and add.
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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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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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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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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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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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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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If \(4,20,100,\ldots\) is a geometric progression, what is the product of the first three terms?
A. (8000)
गुणनफल \(4\cdot20\cdot100=8000\) है। गुणनफल में हर पद को सावधानी से गुणा करें। / The product is \(4\cdot20\cdot100=8000\). Multiply each term carefully in product questions.
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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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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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If (x,18,54) are consecutive terms of a geometric progression, what is (x)?
C. (6)
\(\frac{54}{18}=3\), इसलिए \(\frac{18}{x}=3\) और (x=6)। लगातार अनुपात बराबर रखें। / \(\frac{54}{18}=3\), so \(\frac{18}{x}=3\) and (x=6). Keep consecutive ratios equal.
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What is the (5)th term of the geometric progression \(5,20,80,\ldots\)?
B. (1280)
यहाँ (r=4), इसलिए \(a_5=5\cdot4^4=1280\)। (5)वें पद में (4) बार अनुपात लगता है। / Here (r=4), so \(a_5=5\cdot4^4=1280\). The fifth term uses the ratio (4) times.
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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 \(a_1=\frac{5}{8}\) and (r=2), what is \(a_5\)?
C. (10)
\(a_5=\frac{5}{8}\cdot2^4=10\)। भिन्न पहले पद के साथ घात को सावधानी से गुणा करें। / \(a_5=\frac{5}{8}\cdot2^4=10\). Multiply the power carefully with the fractional first term.
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What is the sum of the first (5) terms of the geometric progression \(3,6,12,24,\ldots\)?
A. (93)
पहले (5) पद (3,6,12,24,48) हैं और योग (93) है। पदों को क्रम से लिखकर जोड़ें। / The first (5) terms are (3,6,12,24,48), and the sum is (93). Write the terms in order and add.
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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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For which (x) will (9,x,49) be consecutive terms of a positive geometric progression?
B. (21)
\(x^2=9\cdot49=441\), इसलिए धनात्मक (x=21)। मध्य पद ज्यामितीय माध्य होता है। / \(x^2=9\cdot49=441\), so positive (x=21). The middle term is the geometric mean.
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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 \(a_2=30\) 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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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 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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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 \(a_n=512\) in \(2,8,32,128,\ldots\), what is (n)?
B. (5)
पद (2,8,32,128,512) हैं, इसलिए (n=5)। पद-संख्या के लिए क्रम से घात पहचानें। / The terms are (2,8,32,128,512), so (n=5). For position identify powers in order.
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In the geometric progression \(4,12,36,108,\ldots\), what is \(a_2+a_4\)?
C. (120)
\(a_2=12\) और \(a_4=108\), इसलिए योग (120) है। दोनों पद अलग निकालकर जोड़ें। / \(a_2=12\) and \(a_4=108\), so the sum is (120). Find both terms separately and then add.
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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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If a geometric progression has \(a_2=18\) and (r=2), what will \(a_5\) be?
C. (144)
\(a_5=a_2r^3=18\cdot2^3=144\)। दिए हुए पद से आगे जाने के लिए अनुपात की सही घात लगाएं। / \(a_5=a_2r^3=18\cdot2^3=144\). To move forward from a given term, use the correct power of the ratio.
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For which (x) will (5,x,45) be consecutive terms of a positive geometric progression?
B. (15)
\(x^2=5\cdot45=225\), इसलिए धनात्मक (x=15)। मध्य पद ज्यामितीय माध्य होता है। / \(x^2=5\cdot45=225\), so positive (x=15). The middle term is the geometric mean.
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What is the average of the first (4) terms of the geometric progression \(15,30,60,120,\ldots\)?
C. (55)
पहले चार पदों का योग (15+30+60+120=225) है और औसत \(\frac{225}{4}\) है, इसलिए दिए विकल्पों में कोई सही नहीं है। औसत के प्रश्न में पहले योग निकालें। / The sum of the first four terms is (15+30+60+120=225), and the average is \(\frac{225}{4}\), so none of the given options is correct. In average questions, first find the sum.
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