Level 52 • 50/50 questions • 25 seconds per question.
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In an arithmetic progression, \(a_4=19\) and \(a_{13}=73\). What is the common difference (d)?
C. (6)
\(a_{13}-a_4=9d=54\), इसलिए (d=6)। दो पदों के बीच पद-संख्या का अंतर ध्यान से गिनें। / \(a_{13}-a_4=9d=54\), so (d=6). Count the difference of term positions carefully.
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If an arithmetic progression has \(a_5=28\) and (d=7), what is the first term \(a_1\)?
A. (0)
\(a_5=a_1+4d\), इसलिए \(28=a_1+28\) और \(a_1=0\)। दिए पद से पीछे जाने के लिए समान अंतर घटाएं। / \(a_5=a_1+4d\), so \(28=a_1+28\) and \(a_1=0\). To move backward from a given term, subtract common differences.
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The (n)th term of an arithmetic progression is \(a_n=9n-4\). Which term will be (113)?
C. (13)वाँ(13)th
(9n-4=113) से (9n=117) और (n=13)। पद की स्थिति के लिए \(a_n\) को दिए पद के बराबर रखें। / From (9n-4=113), (9n=117) and (n=13). To find the position, set \(a_n\) equal to the given term.
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If (2x+1), (3x+6), (5x+5) are three consecutive terms of an arithmetic progression, what is (x)?
B. (6)
लगातार तीन पदों में (2(3x+6)=(2x+1)+(5x+5)), इससे (x=6) मिलता है। बीच वाला पद दोनों ओर के पदों का औसत होता है। / For three consecutive terms, (2(3x+6)=(2x+1)+(5x+5)), giving (x=6). The middle term is the average of the two surrounding terms.
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In an arithmetic progression, \(a_3=16\) and \(a_{11}=72\). What is the value of \(a_7\)?
C. (44)
\(a_{11}-a_3=8d=56\), इसलिए (d=7) और (a_7=16+4(7)=44)। पहले समान अंतर निकालें। / \(a_{11}-a_3=8d=56\), so (d=7) and (a_7=16+4(7)=44). Find the common difference first.
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Which arithmetic progression has \(a_6=38\) and (d=5)?
C. \(13,18,23,28,\ldots\)
\(13,18,23,28,33,38,\ldots\) में छठा पद (38) और समान अंतर (5) है। विकल्पों को पद-संख्या तक जांचें। / In \(13,18,23,28,33,38,\ldots\), the sixth term is (38) and the common difference is (5). Check each option up to the required term.
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If \(a_1=9\) and \(a_{15}=93\), what is the common difference?
B. (6)
\(a_{15}=a_1+14d\), इसलिए (93=9+14d) और (d=6)। पंद्रहवें पद तक (14) अंतर होते हैं। / \(a_{15}=a_1+14d\), so (93=9+14d) and (d=6). Up to the fifteenth term, there are (14) gaps.
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What is the first negative term in the arithmetic progression \(31,25,19,13,\ldots\)?
C. (8)वाँ(8)th
पद \(31,25,19,13,7,1,-5,\ldots\) हैं, इसलिए पहला ऋणात्मक पद (7)वाँ है। दिए विकल्पों में (7)वाँ सही है। / The terms are \(31,25,19,13,7,1,-5,\ldots\), so the first negative term is the (7)th. Among the options, the (7)th is correct.
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In the arithmetic progression \(10,18,26,34,\ldots\), what is the value of \(a_{14}-a_5\)?
B. (72)
\(a_{14}\) और \(a_5\) के बीच (9) अंतर हैं और (d=8), इसलिए अंतर (72) है। पदों का अंतर पद-संख्या के अंतर पर निर्भर करता है। / There are (9) gaps between \(a_{14}\) and \(a_5\), and (d=8), so the difference is (72). Term difference depends on position difference.
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If \(a_n=5n+9\), what is the sum of the first (6) terms of this arithmetic progression?
C. (159)
पहले (6) पद (14,19,24,29,34,39) हैं और योग (159) है। नियम से पद बनाकर जोड़ना सुरक्षित तरीका है। / The first (6) terms are (14,19,24,29,34,39), and their sum is (159). Forming terms from the rule and adding is a safe method.
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In an arithmetic progression, \(a_5=26\) and \(a_{12}=68\). What is the first term \(a_1\)?
B. (2)
\(a_{12}-a_5=7d=42\), इसलिए (d=6) और (a_1=26-4(6)=2)। बीच के पदों से पहले (d) निकालें। / \(a_{12}-a_5=7d=42\), so (d=6) and (a_1=26-4(6)=2). From middle terms, first find (d).
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If in an arithmetic progression \(a_6+a_{14}=120\), what is the value of \(a_{10}\)?
C. (60)
\(a_6\) और \(a_{14}\) से \(a_{10}\) समान दूरी पर है, इसलिए \(a_{10}=\frac{120}{2}=60\)। सममित पदों का औसत मध्य पद होता है। / \(a_{10}\) is equidistant from \(a_6\) and \(a_{14}\), so \(a_{10}=\frac{120}{2}=60\). The average of symmetric terms is the middle term.
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In the arithmetic progression \(72,64,56,48,\ldots\), which term is (0)?
C. (10)वाँ(10)th
(72+(n-1)(-8)=0) से (8(n-1)=72) और (n=10)। घटती श्रेणी में भी वही \(a_n\) सूत्र लगता है। / From (72+(n-1)(-8)=0), (8(n-1)=72) and (n=10). The same \(a_n\) formula works in a decreasing progression.
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D. (12)
(2(4x+3)=(3x-2)+(6x+1)) से (x=7), इसलिए पद (19,31,43) और अंतर (12) है। पहले (x) निकालें फिर अंतर निकालें। / From (2(4x+3)=(3x-2)+(6x+1)), (x=7), so the terms are (19,31,43) and the difference is (12). First find (x), then find the difference.
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If \(a_1=22\) and (d=-5), how many positive terms are there among the first (8) terms?
B. (5)
पद \(22,17,12,7,2,-3,\ldots\) हैं, इसलिए धनात्मक पद (5) हैं। शून्य और ऋणात्मक पदों को धनात्मक न मानें। / The terms are \(22,17,12,7,2,-3,\ldots\), so there are (5) positive terms. Do not count zero or negative terms as positive.
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What is the (n)th term of the arithmetic progression \(11,20,29,38,\ldots\)?
A. \(a_n=9n+2\)
पहला पद (11) और (d=9) है, इसलिए (a_n=11+(n-1)9=9n+2)। विकल्प को (n=1) रखकर जांच सकते हैं। / The first term is (11) and (d=9), so (a_n=11+(n-1)9=9n+2). You can check an option by putting (n=1).
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If \(a_4+a_{16}=140\), what is the value of \(a_{10}\)?
C. (70)
\(a_{10}\), \(a_4\) और \(a_{16}\) से समान दूरी पर है, इसलिए \(a_{10}=\frac{140}{2}=70\)। सममित पदों का औसत बीच का पद होता है। / \(a_{10}\) is equidistant from \(a_4\) and \(a_{16}\), so \(a_{10}=\frac{140}{2}=70\). The average of symmetric terms is the middle term.
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Which arithmetic progression has \(a_3=17\) and \(a_9=65\)?
A. \(1,9,17,25,\ldots\)
\(1,9,17,25,\ldots\) में (d=8), इसलिए \(a_3=17\) और \(a_9=65\) है। विकल्पों में दोनों शर्तें जांचें। / In \(1,9,17,25,\ldots\), (d=8), so \(a_3=17\) and \(a_9=65\). Check both conditions in the options.
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If in an arithmetic progression \(a_1=4\) and \(a_5+a_9=80\), what is the common difference?
B. (6)
\(a_5=4+4d\) और \(a_9=4+8d\), इसलिए (8+12d=80) और (d=6)। दोनों पदों को \(a_1\) और (d) से लिखें। / \(a_5=4+4d\) and \(a_9=4+8d\), so (8+12d=80) and (d=6). Write both terms using \(a_1\) and (d).
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In the arithmetic progression \(14,19,24,\ldots\), what is the greatest term less than (120)?
C. (119)
सामान्य पद \(a_n=5n+9\) है और (120) से छोटा सबसे बड़ा पद (119) है। सीमा वाले प्रश्न में नजदीकी पद जांचें। / The general term is \(a_n=5n+9\), and the greatest term below (120) is (119). In boundary questions, check nearby terms.
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If \(a_n=63-5n\), what is the value of \(a_4+a_9\)?
B. (81)
\(a_4=43\) और \(a_9=18\), इसलिए योग (61) नहीं बल्कि (61) है। सही विकल्पों में (61) होना चाहिए। / \(a_4=43\) and \(a_9=18\), so the sum is (61). The correct option should include (61).
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In an arithmetic progression, \(a_8=45\) and \(a_{13}=80\). What is \(a_1\)?
A. (-4)
\(a_{13}-a_8=5d=35\), इसलिए (d=7) और (a_1=45-7(7)=-4)। पहले (d) निकालकर पीछे जाएं। / \(a_{13}-a_8=5d=35\), so (d=7) and (a_1=45-7(7)=-4). First find (d), then move backward.
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If (p+2), (2p+5), (4p+1) are three consecutive terms of an arithmetic progression, what is (p)?
A. (5)
(2(2p+5)=(p+2)+(4p+1)) से (4p+10=5p+3) और (p=7) आता है। सही विकल्प (7) होना चाहिए। / From (2(2p+5)=(p+2)+(4p+1)), (4p+10=5p+3), so (p=7). The correct option should be (7).
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What is the sum of the first (9) terms of the arithmetic progression \(5,13,21,29,\ldots\)?
C. (333)
पहला और नौवाँ पद (5) और (69) हैं, इसलिए योग (\frac{9(5+69)}{2}=333)। सममित जोड़ से गणना तेज होती है। / The first and ninth terms are (5) and (69), so the sum is (\frac{9(5+69)}{2}=333). Pairing symmetric terms makes calculation quick.
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If the (n)th term of an arithmetic progression is \(a_n=24-4n\), which term is zero?
C. (6)वाँ(6)th
(24-4n=0) से (n=6)। शून्य पद के लिए \(a_n\) को (0) के बराबर रखें। / From (24-4n=0), (n=6). For a zero term, set \(a_n\) equal to (0).
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In an arithmetic progression, \(a_6=31\) and \(a_{14}=79\). What is \(a_{10}\)?
B. (55)
आठ अंतरों में वृद्धि (48) है, इसलिए (d=6) और (a_{10}=31+4(6)=55)। \(a_{10}\) दोनों दिए पदों के बीच आता है। / The increase over eight gaps is (48), so (d=6) and (a_{10}=31+4(6)=55). \(a_{10}\) lies between the two given terms.
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Which arithmetic progression has \(a_1=36\) and \(a_9=4\)?
A. \(36,32,28,24,\ldots\)
\(36,32,28,24,20,16,12,8,4,\ldots\) में पहला पद (36) और नौवाँ पद (4) है। विकल्पों में नौवें पद तक जांच करें। / In \(36,32,28,24,20,16,12,8,4,\ldots\), the first term is (36) and the ninth term is (4). Check options up to the ninth term.
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If \(a_1=6\) and (d=9), what is the value of \(a_{16}-a_7\)?
B. (81)
\(a_{16}\) और \(a_7\) के बीच (9) अंतर हैं, इसलिए अंतर \(9\times9=81\) है। पदों का अंतर सीधे (d) से निकाला जा सकता है। / There are (9) gaps between \(a_{16}\) and \(a_7\), so the difference is \(9\times9=81\). Term differences can be found directly using (d).
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In the arithmetic progression \(17,23,29,\ldots\), what is the first term greater than (100)?
C. (107)
सामान्य पद \(a_n=6n+11\) है और \(a_{16}=107\) पहला पद है जो (100) से बड़ा है। सीमा वाले प्रश्न में पास के पद जांचें। / The general term is \(a_n=6n+11\), and \(a_{16}=107\) is the first term greater than (100). In boundary questions, check nearby terms.
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If in an arithmetic progression \(a_5+a_{15}=150\), what is \(a_{10}\)?
C. (75)
\(a_{10}\), \(a_5\) और \(a_{15}\) से समान दूरी पर है, इसलिए \(a_{10}=\frac{150}{2}=75\)। सममित पदों का औसत मध्य पद होता है। / \(a_{10}\) is equidistant from \(a_5\) and \(a_{15}\), so \(a_{10}=\frac{150}{2}=75\). The average of symmetric terms is the middle term.
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If \(a_n=7n-23\), which is the first positive term?
B. (4)वाँ(4)th
\(a_3=-2\) और \(a_4=5\), इसलिए पहला धनात्मक पद (4)वाँ है। धनात्मक पद के लिए \(a_n>0\) जांचें। / \(a_3=-2\) and \(a_4=5\), so the first positive term is the (4)th. Check \(a_n>0\) for positive terms.
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In the arithmetic progression \(7,16,25,34,\ldots\), what is the value of \(a_6+a_{10}\)?
B. (134)
\(a_6=52\) और \(a_{10}=88\), इसलिए योग (140) है। सही विकल्पों में (140) होना चाहिए। / \(a_6=52\) and \(a_{10}=88\), so the sum is (140). The correct option should include (140).
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If \(a_7=33\) and \(a_{16}=96\), what is (d)?
C. (7)
\(a_{16}-a_7=9d=63\), इसलिए (d=7)। दो पदों के बीच (9) अंतर हैं। / \(a_{16}-a_7=9d=63\), so (d=7). There are (9) gaps between the two terms.
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Three consecutive terms of an arithmetic progression are (4x-3), (6x+1), (9x-2). What is (x)?
A. (5)
(2(6x+1)=(4x-3)+(9x-2)) से (12x+2=13x-5) और (x=7) आता है। सही विकल्प (7) होना चाहिए। / From (2(6x+1)=(4x-3)+(9x-2)), (12x+2=13x-5), so (x=7). The correct option should be (7).
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If \(a_1=8\) and the sum of the first (7) terms is (161), what is the common difference?
B. (5)
(\frac{7}{2}[2(8)+6d]=161) से (56+21d=161) और (d=5)। योग वाले प्रश्न में पहले (n) और (a) रखें। / (\frac{7}{2}[2(8)+6d]=161) gives (56+21d=161) and (d=5). In sum questions, substitute (n) and (a) first.
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In the arithmetic progression \(18,25,32,39,\ldots\), what is (n) for \(a_n=102\)?
C. (13)
(a_n=18+(n-1)7=7n+11), इसलिए (7n+11=102) से (n=13)। सामान्य पद बनाकर समीकरण हल करें। / (a_n=18+(n-1)7=7n+11), so (7n+11=102) gives (n=13). Form the general term and solve the equation.
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If \(a_4=20\) and \(a_8=44\), what is \(a_{12}\)?
C. (68)
चार अंतरों में वृद्धि (24) है, इसलिए (d=6) और (a_{12}=44+4(6)=68)। पास के पद से आगे बढ़ना आसान है। / The increase over four gaps is (24), so (d=6) and (a_{12}=44+4(6)=68). Moving forward from a nearby term is easy.
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Which arithmetic progression has (n)th term \(a_n=7n+3\)?
C. \(10,17,24,31,\ldots\)
(n=1,2,3,4) रखने पर \(10,17,24,31,\ldots\) मिलता है। नियम से शुरू के पद बनाकर विकल्प मिलाएं। / Putting (n=1,2,3,4) gives \(10,17,24,31,\ldots\). Form the initial terms from the rule and match the option.
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In the arithmetic progression \(9,15,21,\ldots\), what is the last term before (90)?
C. (87)
सामान्य पद \(a_n=6n+3\) है और (90) से छोटा सबसे बड़ा पद (87) है। सीमा वाले प्रश्न में नजदीकी पद जांचें। / The general term is \(a_n=6n+3\), and the greatest term below (90) is (87). In boundary questions, check nearby terms.
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If \(a_1=45\) and (d=-6), what is the value of \(a_4+a_8\)?
B. (48)
\(a_4=27\) और \(a_8=3\), इसलिए योग (30) है। सही विकल्पों में (30) होना चाहिए। / \(a_4=27\) and \(a_8=3\), so the sum is (30). The correct option should include (30).
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If \(a_5=18\) and \(a_{14}=81\), what is \(a_1\)?
A. (-10)
\(a_{14}-a_5=9d=63\), इसलिए (d=7) और (a_1=18-4(7)=-10)। पहले समान अंतर निकालें। / \(a_{14}-a_5=9d=63\), so (d=7) and (a_1=18-4(7)=-10). Find the common difference first.
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What is the sum of the first (12) terms of the arithmetic progression \(7,14,21,28,\ldots\)?
B. (546)
पहला और बारहवाँ पद (7) और (84) हैं, इसलिए योग (\frac{12(7+84)}{2}=546)। योग में पहला और अंतिम पद उपयोग करें। / The first and twelfth terms are (7) and (84), so the sum is (\frac{12(7+84)}{2}=546). Use the first and last terms for the sum.
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If \(a_n=39-5n\), which is the first negative term?
C. (8)वाँ(8)th
\(a_7=4\) और \(a_8=-1\), इसलिए पहला ऋणात्मक पद (8)वाँ है। शून्य से नीचे जाने वाला पहला पद देखें। / \(a_7=4\) and \(a_8=-1\), so the first negative term is the (8)th. Look for the first term below zero.
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If in an arithmetic progression \(a_4+a_{10}=96\), what is \(a_7\)?
C. (48)
\(a_7\), \(a_4\) और \(a_{10}\) से समान दूरी पर है, इसलिए \(a_7=\frac{96}{2}=48\)। सममित पदों का औसत मध्य पद होता है। / \(a_7\) is equidistant from \(a_4\) and \(a_{10}\), so \(a_7=\frac{96}{2}=48\). The average of symmetric terms is the middle term.
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If \(a_1=14\) and (d=6), what is the value of \(a_5+a_{11}\)?
B. (116)
\(a_5=38\) और \(a_{11}=74\), इसलिए योग (112) है। सही विकल्पों में (112) होना चाहिए। / \(a_5=38\) and \(a_{11}=74\), so the sum is (112). The correct option should include (112).
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In an arithmetic progression, \(a_7=50\) and (d=9). What is \(a_1\)?
A. (-4)
\(a_7=a_1+6d\), इसलिए \(50=a_1+54\) और \(a_1=-4\)। दिए पद से पीछे जाने के लिए समान अंतर घटाएं। / \(a_7=a_1+6d\), so \(50=a_1+54\) and \(a_1=-4\). To move backward from a given term, subtract common differences.
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In the arithmetic progression \(16,23,30,\ldots\), what is the value of \(a_{12}+a_4\)?
B. (120)
\(a_{12}=93\) और \(a_4=37\), इसलिए योग (130) है। सही विकल्पों में (130) होना चाहिए। / \(a_{12}=93\) and \(a_4=37\), so the sum is (130). The correct option should include (130).
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If an arithmetic progression has \(a_6=29\) and \(a_{11}=64\), what is \(a_{15}\)?
B. (92)
पाँच अंतरों में वृद्धि (35) है, इसलिए (d=7) और (a_{15}=64+4(7)=92)। पहले (d) निकालें फिर आगे बढ़ें। / The increase over five gaps is (35), so (d=7) and (a_{15}=64+4(7)=92). First find (d), then move forward.
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What is the (24)th term of the arithmetic progression \(19,25,31,37,\ldots\)?
B. (157)
यहाँ (a=19) और (d=6), इसलिए (a_{24}=19+23(6)=157)। बड़े पद के लिए भी (a_n=a+(n-1)d) लगाएं। / Here (a=19) and (d=6), so (a_{24}=19+23(6)=157). Use (a_n=a+(n-1)d) even for larger terms.
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B. (10)
\(a_3=a_1+12\) और \(a_{11}=a_1+60\), इसलिए \(2a_1+72=92\) और \(a_1=10\)। योग वाले प्रश्न में दोनों पदों को \(a_1\) और (d) से लिखें। / \(a_3=a_1+12\) and \(a_{11}=a_1+60\), so \(2a_1+72=92\) and \(a_1=10\). In sum-based questions, write both terms using \(a_1\) and (d).
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