Explicit or general rule

Set the explicit formula equal to the target value: n(n + 1)/2 = 28. Multiplying both sides by 2 gives n(n + 1) = 56. We need two consecutive positive integers whose product is 56; they are 7 and 8. Hence n = 7, and direct substitution confirms it: a₇ = 7(7 + 1)/2 = 7×8/2 = 28. Therefore, option D is correct. Checking the distractors gives a₅ = 5×6/2 = 15, a₆ = 6×7/2 = 21, and a₈ = 8×9/2 = 36. These values show why the nearby indices 5, 6, and 8 do not work. The formula represents triangular numbers, and 28 is the seventh triangular number.

Substitute \(n=4\) in the general term: \(a_4=2^4+4=16+4=20\). Therefore, 20 is the correct option. A value such as 18 can result from an incorrect addition after evaluating \(2^4\). Exam tip: evaluate the exponent first, then perform the remaining operations.

Substituting \(n=1,2,3,4\) in \(a_n=2^n+n\) gives \(3,6,11,20\), respectively. Hence, the correct general term is \(a_n=2^n+n\). Although \(a_n=n^2+2\) matches the first three terms, it gives \(18\), not \(20\), when \(n=4\). Exam tip: verify a proposed rule using at least four given terms.

Given \(a_n=3^n-n\). Substituting \(n=3\), we get \(a_3=3^3-3=27-3=24\). Hence, the correct answer is 24. The option 27 is only the value of \(3^3\); subtracting 3 is also required. Exam tip: substitute the value of \(n\) carefully in every part of the general term before calculating.

Evaluate each rule using the term number n, beginning with n = 1. For option D, a₁ = 3¹ − 1 = 2, a₂ = 3² − 2 = 7, a₃ = 3³ − 3 = 24, and a₄ = 3⁴ − 4 = 77. It therefore matches every listed term, so option D is correct. The sequence grows mainly by powers of 3, with the index subtracted. Option A gives 4, 10, 28, 82; option B gives 3, 6, 11, 20; and option C gives 2, 5, 8, 11. Although option A may look similar because it uses a power of 3, its added constant does not fit the second term. Substitution at several indices is the safest verification method for an explicit rule.

Putting (n=5) in (2n^2+2n=60) gives (60). In exams, you can also check quickly by substituting options.

The first differences are 8, 12, and 16, so the sequence is not arithmetic. The second differences are 4 and 4, which suggests a quadratic rule. Substitute n = 1, 2, 3, and 4 into option B: a₁ = 2(1)² + 2(1) = 4; a₂ = 2(2)² + 2(2) = 8 + 4 = 12; a₃ = 18 + 6 = 24; and a₄ = 32 + 8 = 40. Thus option B reproduces the complete given sequence. Option A gives 4, 8, 12, 16 and is linear. Option C gives 4, 10, 18, 28, while option D gives 2, 8, 18, 32. Both fail after the first term, so B is the unique correct general term.

For \(a_n=3n+2\), increasing \(n\) by 1 increases every term by 3, so the common difference is constant and it is an AP. In \(n^2+2\), the differences change. Exam tip: check the difference between consecutive terms.

This is an arithmetic sequence with common difference (5). In exams, use first term and difference in (a_n=a+(n-1)d).

For \(1,4,9,16,\ldots\), \(a_n=n^2\), so the rule has degree 2. The first differences are 3, 5, 7, giving a constant second difference of 2. Exam tip: constant second differences indicate a quadratic sequence.

For (n=1,2,3,4), (2n^2+1) gives the given terms. While checking options, match the first four terms.

((-1)^n) gives (-1) for odd (n) and (1) for even (n). In exams, check odd-even positions in alternating rules.

This is an arithmetic progression because the difference between consecutive terms is 5. Here, \(a=-2\), \(d=5\), and \(n=20\). Therefore, \(a_{20}=a+(n-1)d=-2+(20-1)\times5=-2+95=93\). Hence, 93 is correct. Getting 98 usually results from miscounting the terms or using an incorrect value in place of \(n-1\). Exam tip: for an arithmetic progression, always use \(a_n=a+(n-1)d\).

The common difference is \(d=\frac{a_8-a_3}{8-3}=\frac{39-14}{5}=5\). Hence, \(a_n=a_3+(n-3)d=14+5(n-3)=5n-1\). Therefore, \(5n-1\) is correct. Although \(4n+2\) gives \(a_3=14\), it gives \(a_8=34\), not 39. Exam tip: First find \(d=\frac{a_q-a_p}{q-p}\) from two known terms, then substitute either known term.

The terms are \(1^2, 2^2, 3^2, 4^2,\ldots\). Therefore, the \(n\)th term is \(a_n=n^2\). If \(a_n=n^2+1\), the first term would be 2, which does not match the sequence. Exam tip: write the first few terms alongside \(n=1,2,3,\ldots\) to check the pattern.

The governing concept is substitution into an explicit, or general, rule for a sequence. The formula gives each term directly from its position n. For n=10, a_10=10^2+10+3=100+10+3=113. For n=9, a_9=9^2+9+3=81+9+3=93. Therefore a_10-a_9=113-93=20, so option C is correct. A useful check is to simplify the difference algebraically: (10^2+10+3)-(9^2+9+3)=100+10-81-9=20. The other options result from an arithmetic or substitution error, such as forgetting one term or using an incorrect square.

The governing concept is finding an explicit rule that produces the nth term of a sequence. Test option B at the known positions: when n=1, 1²+2(1)+2=5; when n=2, 2²+2(2)+2=10, which does not equal 11. Thus the supplied answer and option set contain an inconsistency: no listed option reproduces all four terms. In fact, the successive differences are 6, 8, and 10, so the second difference is 2. A quadratic rule has the form n²+bn+c; using the first terms gives b+c=4 and 2b+c=7, hence b=3 and c=1. The correct rule should be a_n=n²+3n+1, absent from the options. This item must be revised before being used.

Given \(a_n=2^n+1\), substitute \(n=6\): \(a_6=2^6+1=64+1=65\). Hence, 65 is the correct option. The value 63 would result from \(2^n-1\), so it is not correct here. Exam tip: In a general term of a sequence, substitute the given value of \(n\) first and then evaluate the exponent.

This is an arithmetic progression with first term \(a=10\) and common difference \(d=7-10=-3\). Therefore, the \(n\)th term is \(a_n=a+(n-1)d=10+(n-1)(-3)=13-3n\). In \(10-3n\), putting \(n=1\) gives 7 rather than the first term 10, so it is incorrect. Exam tip: verify a general term by substituting \(n=1\); it must give the first term.

The governing concept is using an explicit sequence rule to identify the position of a specified term. Set the general term equal to the required value: 6n−4=62. Add 4 to both sides to obtain 6n=66, and divide by 6 to obtain n=11. Hence 62 is the 11th term, so option C is correct. Verification is immediate: a_11=6(11)−4=66−4=62. The other choices would produce a_9=50, a_10=56, and a_12=68, respectively, so they cannot equal 62. The important method is to solve for the index n rather than substitute the given value into the wrong part of the formula.