Sequences

The successive differences are 5, 9, 13, and 17, and each difference increases by 4. Hence, the next difference is \(17+4=21\). Therefore, the next term is \(48+21=69\). Choosing 70 would require the next difference to be 22, which does not follow the pattern. Exam tip: For such sequences, first list consecutive differences and then identify their pattern.

The next difference is (15), so the sixth term is (47+15=62). Exam tip: write changing differences separately.

This is a triangular sequence with (T_n=\frac{n(n+1)}{2}). Since (\frac{9\times10}{2}=45), it is the (9)th term.

This is a geometric sequence with first term 2 and common ratio 2. Therefore, \(a_8=2\times 2^{7}=2^{8}=256\). The value 128 is the seventh term, so it is a close but incorrect option. Exam tip: write terms of a doubling sequence as powers of 2.

Given \(a_n=n^2+3\) and \(a_n=84\), we get \(n^2+3=84\). Thus, \(n^2=81\), so \(n=9\). Therefore, 84 is the 9th term of the sequence. The 8th term is \(8^2+3=67\), so it is not correct. Exam tip: to find a term number, substitute the given term value in the formula for \(a_n\) and solve for \(n\).

The differences between consecutive terms are \(13-6=7\), \(24-13=11\), \(39-24=15\), and \(58-39=19\). These differences increase by \(4\) each time, so the next difference is \(23\). Hence, the next term is \(58+23=81\). Choosing \(80\) would give a difference of \(22\), which does not follow the pattern. Exam tip: For such sequences, first list consecutive differences and look for their pattern.

The nth term is n^3. Therefore, the 6th term is 6^3 = 216 and the 5th term is 5^3 = 125. Hence, the difference is 216 - 125 = 91. Note that 81 is 4^3, not the required difference. Exam tip: evaluate both specified terms first, then subtract the smaller term from the larger one.

Odd positions are (10,11,12,\ldots), so the next odd-position term is (13). Exam tip: separate positions in alternating sequences.

Odd positions are (2,4,6,\ldots), so the next odd-position term is (8). Exam tip: split the two patterns in alternating sequences.

The general term is given as (a_n=n^2+2). Substituting n=9, we get (a_9=9^2+2=81+2=83). The value 81 is only (9^2); the additional 2 must also be included. Exam tip: when a general term is given, substitute the required term number directly for n.

This is an arithmetic sequence with first term 12 and common difference 5. Its nth term is \(a_n=12+5(n-1)\). Using \(a_n<60\), we get \(12+5(n-1)<60\), so \(n<10.6\). Thus, the possible whole-number values of \(n\) are 1 through 10, giving 10 terms less than 60. The 11th term is 62, so it is not included. Exam tip: for “less than,” use the strict inequality \(<\), not \(\leq\).

In option A, the consecutive differences are 5−2=3 and 8−5=3, so the difference is constant. However, the ratios 5/2 and 8/5 differ. In exams, check differences and ratios separately.

This is a geometric sequence with first term \(a=5\) and common ratio \(r=2\). Its \(n\)th term is \(a_n=5\times2^{n-1}\). From \(5\times2^{n-1}=320\), we get \(2^{n-1}=64=2^6\). Therefore, \(n-1=6\) and \(n=7\). Hence, 320 is the 7th term. The 6th term is \(160\), so it is not correct. Exam tip: use \(a_n=ar^{n-1}\) to find a term's position in a geometric sequence.

The sixth term is (16+26=42) and the seventh is (26+42=68). Exam tip: count positions carefully in recursive sequences.

The direct answer is option C, \(a_n=n^2+2n-1\). Test it at n=1: \(1+2-1=2\); at n=2: \(4+4-1=7\); at n=3: \(9+6-1=14\); at n=4: \(16+8-1=23\). These match the sequence, and n=5 gives \(25+10-1=34\). Option A, \(n^2+1\), gives 2, 5, 10, 17, so it fails at the second term. Option B, \(n^2+n\), gives 2, 6, 12, 20, so it also fails; the question correctly warns against it. Option C matches every listed term. Option D, \(2n^2\), gives 2, 8, 18, 32, which does not match. Memory cue: substitute n=1, 2, and 3 into each candidate before choosing a general term.

The direct answer is option A, \(a_n=n^2-1\). Number the terms starting with n=1. Then \(1^2-1=0\), \(2^2-1=3\), \(3^2-1=8\), \(4^2-1=15\), and \(5^2-1=24\), exactly matching the sequence. Option A is therefore correct. Option B, \(n^2+1\), gives 2 as its first term instead of 0. Option C, \(2n-2\), is an arithmetic expression with constant difference 2, while the given differences are 3, 5, 7, and 9. Option D, \(n^2+n\), gives 2, 6, 12, 20, so it does not match. The terms are one less than consecutive squares: \(1^2-1,2^2-1,3^2-1,\ldots\). Memory cue: when differences are consecutive odd numbers, think of squares.

The consecutive differences are 4, 6, 8, and 10, increasing by 2 each time, so the sequence is quadratic in nature. Substituting \(n=1,2,3,4,5\) in \(a_n=n^2+n+1\) gives 3, 7, 13, 21, and 31 respectively. Option C gives 7 when \(n=2\), but it gives 1 rather than 3 when \(n=1\). Exam tip: verify a proposed general term using at least the first three or four terms.

The differences between consecutive terms are 18−11=7, 27−18=9, 38−27=11, and 51−38=13. Each difference increases by 2, so the next difference is 15. Hence, the next term is 51+15=66. Choosing 64 would mean adding 13 again, which does not follow the increasing-difference pattern. Exam tip: For such sequences, first list the consecutive differences and check their pattern.

This question has inconsistent options because (n^3+n) gives (10) at (n=2), not (12). Exam tip: verify every option before accepting a pattern.

(n(n+1)^2) does not give the full sequence, so care is needed. None of the given options matches all terms.

This is (3^2,4^2,5^2,\ldots), so the (20)th term is (22^2=484). Exam tip: track how the index shifts.

This is (2^3,3^3,4^3,\ldots), so the (10)th term is (11^3=1331). Exam tip: handle shifted cube sequences carefully.

The rule is: next term = twice the previous term − 1. Therefore, the term after 33 is \(2\times 33-1=66-1=65\). Getting 63 would mean subtracting 3 from 66, but the rule requires subtracting only 1. Exam tip: in a recursive sequence, apply the stated rule directly to the last given term.

The rule is: next term = twice the previous term + 1. Therefore, the term after 47 is \(2\times47+1=94+1=95\). Option 94 is only twice 47; it misses the required addition of 1. Exam tip: apply the stated rule directly to the last given term to verify the answer.

After (10.5), dividing by (2) gives (5.25). Exam tip: identify the next operation in alternating-operation sequences.