nth term

Correct answer: A. Arithmetic progression

Explanation: This is an arithmetic progression because \(a_{n+1}-a_n=[6(n+1)+1]-(6n+1)=6\), a constant difference. A geometric progression requires a constant ratio instead. Exam tip: check consecutive-term differences first.

Correct answer: A. (a_n=6n)

Explanation: The direct answer is option A, \(a_n=6n\). Look at the terms: 6, 12, 18, and 24. They are respectively \(6\times1\), \(6\times2\), \(6\times3\), and \(6\times4\). Therefore the nth term is \(6\times n=6n\). This is also an arithmetic progression with first term 6 and common difference 6: \(a_n=6+(n-1)6=6n\). Option A is correct because it gives 6 for \(n=1\), 12 for \(n=2\), and so on. Option B, \(n+6\), gives 7 when \(n=1\), so it does not even give the first term. Option C, \(12n\), gives 12 for the first term, which is too large. Option D, \(6n-1\), gives 5 for the first term and therefore fails. The memory trick is: when the sequence is 6 times 1, 2, 3, 4, its nth term is 6 times n.

Correct answer: B. 34

Explanation: For the sixth term, substitute \(n=6\). Thus, \(a_6=6^2-2=36-2=34\). Therefore, 34 is correct. The close distractor 36 is only \(6^2\); the subtraction of 2 has not been performed. Exam tip: after substituting the value of \(n\), evaluate the power first and then subtract.

Correct answer: B. aₙ = 3n − 2

Explanation: Direct answer: aₙ = 3n − 2, so B is correct. The differences are constant: 4 − 1 = 3, 7 − 4 = 3, and 10 − 7 = 3. Therefore the sequence is arithmetic, with first term a₁ = 1 and common difference d = 3. Use aₙ = a₁ + (n − 1)d. Substitution gives aₙ = 1 + (n − 1)3 = 1 + 3n − 3 = 3n − 2. Verify it at several positions: n = 1 gives 1, n = 2 gives 4, n = 3 gives 7, and n = 4 gives 10. A starts at 4, so it is incorrect. C has difference 1, not 3. D starts at 1 but has difference 4, so it matches neither the second term nor the pattern. Common confusion: the coefficient of n is the common difference, but the constant term must be adjusted so that n = 1 gives the actual first term.

Correct answer: A. एक अंकगणितीय अनुक्रम, जिसका सार्वांतर 5 है

Explanation: In \(a_n=5n+2\), the coefficient of \(n\) is 5, so each successive term increases by 5. Hence it is an arithmetic sequence with common difference 5. A geometric sequence requires a constant ratio between consecutive terms. Exam tip: in \(a_n=dn+c\), identify \(d\) as the common difference.

Correct answer: C. tenth term

Explanation: This sequence consists of multiples of 7, so its nth term is \(a_n=7n\). Putting \(7n=70\) gives \(n=10\). Therefore, 70 is the tenth term. The ninth term is 63, so option B is not correct. Exam tip: for this sequence, divide the given number by 7 to find its term number.

Correct answer: B. 5

Explanation: For the seventh term, substitute 7 for n: \(a_7=12-7=5\). Therefore, the correct answer is 5. Getting 6 would result from subtracting the wrong term number from 12. Exam tip: to find the nth term, replace n with the required term number.

Correct answer: D. 17

Explanation: This is an arithmetic sequence because 3 is added to each successive term. Starting with 2, the sixth term is obtained by adding 3 five times: \(2+5\times3=17\). Therefore, 17 is correct. The value 16 would result from adding 3 only four times. Exam tip: to find the \(n\)th term, add the common difference \(n-1\) times to the first term.

Correct answer: A. \(a_n=3n+1\)

Explanation: This is an arithmetic sequence because each successive term increases by 3. For the first term, \(a_1=3(1)+1=4\), so \(a_n=3n+1\) is correct. The formula \(3n-1\) gives 2 as the first term. Exam tip: verify a formula using the first two terms.

Correct answer: C. \(64\)

Explanation: For the fourth term, substitute \(n=4\). Thus, \(a_4=4^3=4\times4\times4=64\). Therefore, \(64\) is the correct answer. \(16\) is a close distractor because it equals \(4^2\), not \(4^3\). Exam tip: \(n^3\) means multiplying \(n\) by itself three times.

Correct answer: B. 80

Explanation: This is an arithmetic progression with first term \(a=8\) and common difference \(d=8\). Thus, \(a_{10}=a+(10-1)d=8+9\times8=80\). The value 72 is the ninth term, so it is a close but incorrect option. In exams, use \(a_n=a+(n-1)d\) for the nth term of an AP.

Correct answer: A. \(a_n=4n+1\)

Explanation: \(a_n=4n+1\) has the linear form \(pn+q\), so consecutive terms differ by the constant \(4\); hence it is an arithmetic progression. For \(n^2+1\), the differences change. Exam tip: check whether \(a_{n+1}-a_n\) is constant.

Correct answer: A. aₙ = 5n − 2

Explanation: Direct answer: aₙ = 5n − 2, so A is correct. Subtract consecutive terms: 8 − 3 = 5, 13 − 8 = 5, and 18 − 13 = 5. The constant difference is 5, so this is an arithmetic progression. Its first term is a₁ = 3. Using aₙ = a₁ + (n − 1)d, we get aₙ = 3 + (n − 1)5 = 3 + 5n − 5 = 5n − 2. Verify: n = 1 gives 5 − 2 = 3; n = 2 gives 10 − 2 = 8; n = 3 gives 15 − 2 = 13; and n = 4 gives 20 − 2 = 18. B gives 7 at the first position, so it fails immediately. C gives 8 at n = 1 and changes by 3, while D gives 6 at n = 1 and changes by 1. Memory cue: in an arithmetic sequence, start with the first term and add the common difference n − 1 times, not n times.

Correct answer: A. 1, 4, 7, 10, ...

Explanation: Putting \(n=1\) gives the first term \(3(1)-2=1\), and \(n=2\) gives \(3(2)-2=4\). Hence the sequence is 1, 4, 7, 10, .... In exams, verify the first two terms to identify the sequence quickly.

Correct answer: C. ninth term

Explanation: This sequence consists of multiples of 9, so its nth term is \(a_n=9n\). Putting \(9n=81\) gives \(n=9\). Therefore, 81 is the ninth term. The eighth term is \(9\times8=72\), so it is not correct. Exam tip: To find a term number, equate the given value to the nth-term formula and solve for \(n\).

Correct answer: B. 4

Explanation: For the sixth term, substitute \(n=6\) in the rule: \(a_6=\frac{6+2}{2}=\frac{8}{2}=4\). Therefore, the correct answer is 4. Option 3 can result from an incorrect division after adding 6 and 2. Exam tip: in \(a_n\), first substitute the given subscript for \(n\), then simplify step by step.

Correct answer: A. (a_n=\frac{n+1}{2})

Explanation: At (n=1) it gives (1), and at (n=2) it gives (\frac{3}{2}), so the rule is correct. In exams, match fractional sequences with terms.

Correct answer: C. 27

Explanation: For the third term, substitute n=3. Thus, a_3=3^3=27, so 27 is correct. The value 9 equals 3^2, so it is the second term, not the third. Exam tip: To find the nth term, replace n in the formula with the required term number.

Correct answer: B. 34

Explanation: This is an arithmetic progression with first term \(a=4\) and common difference \(d=9-4=5\). Its \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_7=4+(7-1)\times5=34\). Hence, 34 is correct. The value 29 is the fifth term, not the seventh term. Exam tip: identify the first term and common difference before substituting \(n\) in \(a_n=a+(n-1)d\).

Correct answer: A. \(a_n=2n+5\)

Explanation: Twice the position number \(n\) is \(2n\). Adding 5 gives \(a_n=2n+5\). Option \(2n-5\) incorrectly subtracts 5. Exam tip: translate “more than” as addition before choosing the formula.

Correct answer: A. \(a_n=12n\)

Explanation: This is an arithmetic progression with first term \(a=12\) and common difference \(d=12\). Thus, \(a_n=a+(n-1)d=12+(n-1)\times12=12n\). The expression \(a_n=12n+12\) gives 24 as the first term, so it is incorrect. Exam tip: Substitute \(n=1\) to check whether the rule gives the first term.

Correct answer: C. 20

Explanation: For the fourth term, substitute n=4. Thus, a_4=4^2+4=16+4=20. Therefore, 20 is the correct answer. The value 16 is only 4^2; the additional +4 must also be included. Exam tip: when finding an nth term, substitute the term number carefully in the formula.

Correct answer: A. \(a_n=n^2+n\)

Explanation: The rule for this sequence is \(a_n=n(n+1)=n^2+n\). Substituting \(n=1,2,3,4\) gives \(2,6,12,20\), respectively. With \(a_n=n^2\), the second term would be \(4\), not the given \(6\). Exam tip: verify a proposed nth-term rule using the first few terms.

Correct answer: B. The position or number of the term

Explanation: \(n\) indicates the position of a term, so \(a_4\) is the fourth term of the sequence. The value of \(a_n\) comes from its rule; the common difference is a separate quantity. Exam tip: read the subscript as the term number.

Correct answer: C. ninth term

Explanation: Each term in this sequence is a multiple of 10, so its nth term is \(a_n=10n\). Putting \(10n=90\) gives \(n=9\). Therefore, 90 is the ninth term. The tenth term would be 100, so it is not correct. Exam tip: for this sequence, divide the given number by 10 to find its term number quickly.