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The first term is (5) and the common difference is (7), so (a_n=5+(n-1)7=7n-2). In an arithmetic sequence the coefficient is the common difference.
Step 2
Why this answer is correct
The correct answer is B. \(a_n=7n-2\). The first term is (5) and the common difference is (7), so (a_n=5+(n-1)7=7n-2). In an arithmetic sequence the coefficient is the common difference.
Step 3
Exam Tip
पहला पद (5) और समान अंतर (7) है इसलिए (a_n=5+(n-1)7=7n-2)। समानांतर अनुक्रम में गुणांक समान अंतर होता है।
From the given terms, (p+q=5) and (2p+q=7), so (p=2), (q=3), and \(a_4=45\). When coefficients are unknown, first form equations using small terms.
Step 2
Why this answer is correct
The correct answer is C. (45). From the given terms, (p+q=5) and (2p+q=7), so (p=2), (q=3), and \(a_4=45\). When coefficients are unknown, first form equations using small terms.
Step 3
Exam Tip
दिए पदों से (p+q=5) और (2p+q=7), इसलिए (p=2), (q=3) और \(a_4=45\)। अज्ञात गुणांक हों तो पहले छोटे पदों से समीकरण बनाएं।
The first term is (42) and the difference is (-6), so (a_n=42+(n-1)(-6)=48-6n). Keep the difference negative for a decreasing sequence.
Step 2
Why this answer is correct
The correct answer is C. \(a_n=48-6n\). The first term is (42) and the difference is (-6), so (a_n=42+(n-1)(-6)=48-6n). Keep the difference negative for a decreasing sequence.
Step 3
Exam Tip
पहला पद (42) और अंतर (-6) है इसलिए (a_n=42+(n-1)(-6)=48-6n)। घटते अनुक्रम में अंतर ऋणात्मक रखें।
The numerator is (2n+1) and the denominator is (3n+2), so \(a_n=\frac{2n+1}{3n+2}\). In fractions observe numerator and denominator patterns separately.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=\frac{2n+1}{3n+2}\). The numerator is (2n+1) and the denominator is (3n+2), so \(a_n=\frac{2n+1}{3n+2}\). In fractions observe numerator and denominator patterns separately.
Step 3
Exam Tip
अंश (2n+1) और हर (3n+2) है इसलिए \(a_n=\frac{2n+1}{3n+2}\)। भिन्नों में अंश और हर का पैटर्न अलग देखें।
Its rule is \(a_n=3n^2+4n+4\), so the (10)th term is (344), not any listed option. In exams also check the consistency of options.
Step 2
Why this answer is correct
The correct answer is C. (331). Its rule is \(a_n=3n^2+4n+4\), so the (10)th term is (344), not any listed option. In exams also check the consistency of options.
Step 3
Exam Tip
इसका नियम \(a_n=3n^2+4n+4\) है इसलिए \(a_{10}=300+40+4=344\) नहीं बल्कि विकल्पों से मिलान गलत है; सही (10)वाँ पद (344) है। विकल्पों की संगति भी परीक्षा में जांचें।
The general term is \(a_n=8n+5\), and (8n+5=93) gives (n=11). If (n) is a natural number, the given term belongs to the sequence.
Step 2
Why this answer is correct
The correct answer is B. (93) ग्यारहवाँ पद है / (93) is the eleventh term. The general term is \(a_n=8n+5\), and (8n+5=93) gives (n=11). If (n) is a natural number, the given term belongs to the sequence.
Step 3
Exam Tip
सामान्य पद \(a_n=8n+5\) है और (8n+5=93) से (n=11)। यदि (n) प्राकृतिक संख्या हो तो दिया पद अनुक्रम में आता है।
This is \(3^2,7^2,11^2,15^2,\ldots\), so (a_n=(4n-1)2). In square sequences observe the difference between bases.
Step 2
Why this answer is correct
The correct answer is A. (a_n=(4n-1)2). This is \(3^2,7^2,11^2,15^2,\ldots\), so (a_n=(4n-1)2). In square sequences observe the difference between bases.
Step 3
Exam Tip
यह \(3^2,7^2,11^2,15^2,\ldots\) है इसलिए (a_n=(4n-1)2)। वर्गों में आधारों का अंतर देखें।
The numerator is (3n+2) and the denominator is (4n+2), so \(a_n=\frac{3n+2}{4n+2}\). In a fractional sequence form separate rules for both parts.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=\frac{3n+2}{4n+2}\). The numerator is (3n+2) and the denominator is (4n+2), so \(a_n=\frac{3n+2}{4n+2}\). In a fractional sequence form separate rules for both parts.
Step 3
Exam Tip
अंश (3n+2) और हर (4n+2) है इसलिए \(a_n=\frac{3n+2}{4n+2}\)। भिन्न अनुक्रम में दोनों भागों का अलग नियम बनाएं।
The increase over five gaps is (35), so (d=7), then \(a_n=7n-5\). From two given terms first find the common difference.
Step 2
Why this answer is correct
The correct answer is B. \(a_n=7n-5\). The increase over five gaps is (35), so (d=7), then \(a_n=7n-5\). From two given terms first find the common difference.
Step 3
Exam Tip
पाँच अंतरों में वृद्धि (35) है इसलिए (d=7), फिर \(a_n=7n-5\)। दो दिए पदों से पहले समान अंतर निकालें।
The first term is (17) and the difference is (-9), so (a_n=17+(n-1)(-9)=26-9n). Treat decrease as a negative difference.
Step 2
Why this answer is correct
The correct answer is B. \(a_n=26-9n\). The first term is (17) and the difference is (-9), so (a_n=17+(n-1)(-9)=26-9n). Treat decrease as a negative difference.
Step 3
Exam Tip
पहला पद (17) और अंतर (-9) है इसलिए (a_n=17+(n-1)(-9)=26-9n)। घटने को ऋणात्मक अंतर मानें।
This is three times the triangular numbers, so (a_n=\frac{3n(n+1)}{2}). Recognize the pattern from additions (6,9,12).
Step 2
Why this answer is correct
The correct answer is A. (a_n=\frac{3n(n+1)}{2}). This is three times the triangular numbers, so (a_n=\frac{3n(n+1)}{2}). Recognize the pattern from additions (6,9,12).
Step 3
Exam Tip
यह (3) गुना त्रिभुज संख्याओं का क्रम है इसलिए (a_n=\frac{3n(n+1)}{2})। लगातार जोड़ (6,9,12) से पैटर्न पहचानें।
This is \(4^2,8^2,12^2,16^2,\ldots\), so (a_n=(4n)2=16n-2). In square sequences with equal base gaps, find the base rule.
Step 2
Why this answer is correct
The correct answer is D. \(a_n=16n^2\). This is \(4^2,8^2,12^2,16^2,\ldots\), so (a_n=(4n)2=16n-2). In square sequences with equal base gaps, find the base rule.
Step 3
Exam Tip
यह \(4^2,8^2,12^2,16^2,\ldots\) है इसलिए (a_n=(4n)2=16n-2)। सम अंतर वाले वर्गों में आधार का नियम देखें।
The magnitude is \(6,11,16,21,\ldots\) and signs start positive and alternate, so (a_n=(-1)^{n+1}(5n+1)). The sign of the first term decides the power.
Step 2
Why this answer is correct
The correct answer is B. (a_n=(-1)^{n+1}(5n+1)). The magnitude is \(6,11,16,21,\ldots\) and signs start positive and alternate, so (a_n=(-1)^{n+1}(5n+1)). The sign of the first term decides the power.
Step 3
Exam Tip
परिमाण \(6,11,16,21,\ldots\) है और चिह्न धन से शुरू होकर बदलता है इसलिए (a_n=(-1)^{n+1}(5n+1))। पहले पद का चिह्न शक्ति तय करता है।
The change over six gaps is (-36), so (d=-6), hence \(a_n=31-6n\). From two given terms first find the common difference.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=31-6n\). The change over six gaps is (-36), so (d=-6), hence \(a_n=31-6n\). From two given terms first find the common difference.
Step 3
Exam Tip
छह अंतरों में परिवर्तन (-36) है इसलिए (d=-6), अतः \(a_n=31-6n\)। दो दिए पदों से पहले समान अंतर निकालें।
(a_4=\frac{4(4)+1}{3(4)-1}=\frac{17}{11}). In a fractional rule substitute (n) in both numerator and denominator.
Step 2
Why this answer is correct
The correct answer is C. \(\frac{17}{11}\). (a_4=\frac{4(4)+1}{3(4)-1}=\frac{17}{11}). In a fractional rule substitute (n) in both numerator and denominator.
Step 3
Exam Tip
(a_4=\frac{4(4)+1}{3(4)-1}=\frac{17}{11})। भिन्न वाले नियम में अंश और हर दोनों में (n) रखें।
In a linear rule the coefficient of (n) is (-5), so the common difference is (-5). Do not miss the sign while reading the coefficient.
Step 2
Why this answer is correct
The correct answer is A. (-5). In a linear rule the coefficient of (n) is (-5), so the common difference is (-5). Do not miss the sign while reading the coefficient.
Step 3
Exam Tip
रैखिक नियम में (n) का गुणांक (-5) है इसलिए समान अंतर (-5) है। गुणांक पढ़ते समय चिह्न न छोड़ें।
\(a_4=95\) and \(a_2=23\), so the sum is (118). The correct option is (118), and both terms should be found separately.
Step 2
Why this answer is correct
The correct answer is B. (120). \(a_4=95\) and \(a_2=23\), so the sum is (118). The correct option is (118), and both terms should be found separately.
Step 3
Exam Tip
\(a_4=95\) और \(a_2=23\), इसलिए योग (118) है। सही विकल्प (118) है और दोनों पद अलग निकालें।
The general term is \(a_n=79-9n\), and (79-9n=-2) gives (n=9). Even in decreasing sequences the position is natural.
Step 2
Why this answer is correct
The correct answer is C. (9)वाँ / (9)th. The general term is \(a_n=79-9n\), and (79-9n=-2) gives (n=9). Even in decreasing sequences the position is natural.
Step 3
Exam Tip
सामान्य पद \(a_n=79-9n\) है और (79-9n=-2) से (n=9)। घटते अनुक्रम में भी पद-संख्या प्राकृतिक होती है।
The numerator is \(n^2+5\) and the denominator is \(n^2+4\), so \(a_n=\frac{n^2+5}{n^2+4}\). Identify square patterns in fractional sequences.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=\frac{n^2+5}{n^2+4}\). The numerator is \(n^2+5\) and the denominator is \(n^2+4\), so \(a_n=\frac{n^2+5}{n^2+4}\). Identify square patterns in fractional sequences.
Step 3
Exam Tip
अंश \(n^2+5\) और हर \(n^2+4\) है इसलिए \(a_n=\frac{n^2+5}{n^2+4}\)। भिन्न अनुक्रम में वर्ग पैटर्न पहचानें।
The numerator is (3n) and the denominator is (5n+2), so \(a_n=\frac{3n}{5n+2}\). Observe the denominator growth carefully.
Step 2
Why this answer is correct
The correct answer is A. \(a_n=\frac{3n}{5n+2}\). The numerator is (3n) and the denominator is (5n+2), so \(a_n=\frac{3n}{5n+2}\). Observe the denominator growth carefully.
Step 3
Exam Tip
अंश (3n) और हर (5n+2) है इसलिए \(a_n=\frac{3n}{5n+2}\)। हर की बढ़त को ध्यान से देखें।
The magnitude is (7n) and signs start positive and alternate, so (a_n=(-1)^{n+1}7n). Choose the power of ((-1)) by checking the first term sign.
Step 2
Why this answer is correct
The correct answer is D. (a_n=(-1)^{n+1}7n). The magnitude is (7n) and signs start positive and alternate, so (a_n=(-1)^{n+1}7n). Choose the power of ((-1)) by checking the first term sign.
Step 3
Exam Tip
परिमाण (7n) है और चिह्न धन से शुरू होकर बदलता है इसलिए (a_n=(-1)^{n+1}7n)। पहले पद का चिह्न देखकर ((-1)) की शक्ति चुनें।
\(a_5=2\) and \(a_6=-2\), so the first negative term is the (6)th. Do not count zero or positive terms as negative.
Step 2
Why this answer is correct
The correct answer is B. (6)वाँ / (6)th. \(a_5=2\) and \(a_6=-2\), so the first negative term is the (6)th. Do not count zero or positive terms as negative.
Step 3
Exam Tip
\(a_5=2\) और \(a_6=-2\) है, इसलिए पहला ऋणात्मक पद (6)वाँ है। शून्य या धनात्मक पद को ऋणात्मक न मानें।
The increase over six gaps is (54), so (d=9), hence (a_{15}=83+4(9)=119). Extend the terms using the common difference.
Step 2
Why this answer is correct
The correct answer is C. (119). The increase over six gaps is (54), so (d=9), hence (a_{15}=83+4(9)=119). Extend the terms using the common difference.
Step 3
Exam Tip
छह अंतरों में वृद्धि (54) है इसलिए (d=9), अतः (a_{15}=83+4(9)=119)। समान अंतर को आगे बढ़ाकर पद निकालें।
\(n^3+n-1\) does not give the sequence; none of the listed options correctly fits this sequence. Matching options with initial terms is necessary.
Step 2
Why this answer is correct
The correct answer is B. \(a_n=n^3+n-1\). \(n^3+n-1\) does not give the sequence; none of the listed options correctly fits this sequence. Matching options with initial terms is necessary.
Step 3
Exam Tip
\(n^3+n-1\) से (1,9,29,67) नहीं मिलता; इस क्रम के लिए दिए विकल्पों में सही नियम नहीं है। विकल्पों को शुरुआती पदों से मिलाना जरूरी है।
