Explicit or general rule

The governing concept is an explicit or general rule for a sequence. Such a rule gives a term directly from its position, so there is no need to extend the sequence one term at a time. Substitute n=12 into a_n=n²+1: a_12=12²+1=144+1=145. Similarly, a_10=10²+1=100+1=101. Therefore a_12−a_10=145−101=44, making option C correct. A frequent mistake is to subtract the indices and report 12−10=2, but the question asks for the difference between term values. Options A, B, and D can result from squaring or subtracting incorrectly. The explicit formula provides a complete and direct calculation.

The governing concept is an explicit rule, which determines any term directly from its index. For n=15, a_15=2(15²)+2(15)+2=2(225)+30+2=482. For n=13, a_13=2(13²)+2(13)+2=2(169)+26+2=366. Thus a_15−a_13=482−366=116, so option D is correct. It would be incorrect to assume a constant first difference merely from the listed terms, because the supplied rule is quadratic and its differences change. Options A, B, and C may arise from incomplete substitution, using an incorrect square, or making an arithmetic error during subtraction. Applying the given formula to both indices avoids those mistakes.

The governing concept is evaluating an explicit sequence rule while tracking alternating signs. For n = 7, (-1)^7 = -1, so a_7 = -1(7 + 1) = -8. For n = 8, (-1)^8 = 1, so a_8 = 1(8 + 1) = 9. Therefore a_7 + a_8 = -8 + 9 = 1. Hence option B is correct. The sign changes because odd powers of -1 are negative and even powers are positive. Option A would reverse or mishandle one sign, while options C and D arise from adding the magnitudes incorrectly or ignoring the alternating factor. Substitution into the given rule is the safest method.

The governing concept is recognizing a multiplicative sequence generated by factorials. Each term is multiplied by the next positive integer: 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24, and 24 × 5 = 120. The next operation must therefore multiply 120 by 6, giving 120 × 6 = 720. Equivalently, the displayed terms are 1!, 2!, 3!, 4!, and 5!, so the next term is 6! = 720. Hence option B is correct. Option A uses 5 instead of 6; 840 and 960 do not follow the successive-factor pattern. The sequence is not arithmetic or geometric because its differences and ratios are not constant.

The governing concept is locating a term by using an explicit product-form rule. We require n(n + 3) = 130. Directly testing the central option gives n = 10: a_10 = 10(10 + 3) = 10 × 13 = 130. Thus 130 is the 10th term. Algebraically, expanding gives n² + 3n − 130 = 0, which factors as (n + 13)(n − 10) = 0. The roots are n = −13 and n = 10, but only the positive integer can be a sequence position. Checking the other choices gives a_8 = 8 × 11 = 88, a_9 = 9 × 12 = 108, and a_11 = 11 × 14 = 154. None equals 130, so option C is unambiguous.

The governing concept is evaluation of an explicit sequence rule containing an alternating-sign term. Substitute n = 10 and n = 9 separately. Since 10 is even, (−1)^10 = 1, so a_10 = 5(10) + 1 = 51. Since 9 is odd, (−1)^9 = −1, so a_9 = 5(9) − 1 = 44. Therefore, a_10 − a_9 = 51 − 44 = 7, making option C correct. The alternating term cannot be treated as having the same sign for both indices. Option A would result from ignoring the change in sign, while options B and D reflect arithmetic or subtraction errors. Evaluating each term independently avoids these mistakes.

The governing concept is substitution into an explicit, or general, term formula. This type of rule gives any term directly when its position n is known. For the eleventh term, put n = 11 into the expression: a_11 = 4(11^2) + 2(11) − 3. First, 11^2 = 121. Then 4 × 121 = 484 and 2 × 11 = 22, so a_11 = 484 + 22 − 3 = 503. Therefore option B is correct. The answer is not obtained by merely multiplying 11 by the coefficients; the square must be calculated before substitution is completed. Options A, C, and D are plausible results of errors in squaring, multiplication, or handling the constant term.

The governing concepts are explicit-term evaluation and the alternating sign produced by powers of −1. For n=8, the exponent n+1 is 9, so (−1)^9=−1; hence a_8=−(2×8+3)=−19. For n=9, the exponent is 10, so (−1)^10=+1; hence a_9=2×9+3=21. Therefore a_8+a_9=−19+21=2, so option B is correct. Option A incorrectly assumes equal magnitudes that cancel, although the magnitudes are 19 and 21. Option C may come from an arithmetic error, while option D results from ignoring the alternating sign or adding the magnitudes. Evaluating each term separately prevents these mistakes.

The direct answer is option B, \(a_n=5n-1\). The terms increase by 5: \(9-4=5\), \(14-9=5\), and \(19-14=5\). Thus this is an arithmetic sequence with first term 4 and common difference 5. Using \(a_n=a_1+(n-1)d\), we get \(a_n=4+(n-1)5=4+5n-5=5n-1\). Option A gives \(a_1=6\), so it cannot be correct. Option B gives 4, 9, 14, and 19 for \(n=1,2,3,4\), exactly as required. Option C gives \(a_1=9\) and increases by 4, so both its starting value and difference are wrong. Option D gives 5, 6, 7, 8, increasing by only 1. The safest exam method is to check the first term and the common difference.

In \(a_n=5n+2\), the coefficient of \(n\) is constant, so consecutive terms always differ by 5. Hence it is an arithmetic progression. Rules involving \(n^2\) or \(2^n\) do not have a constant difference. Exam tip: a rule of the form \(an+b\) represents an AP.

The difference is (7), and at (n=1) the value must be (12), so (a_n=7n+5). In exams, take the difference as the coefficient of (n).

Given \(a_n=50-4n\), substitute \(n=9\): \(a_9=50-4(9)=50-36=14\). Therefore, 14 is correct. A value such as 16 can result from incorrectly evaluating \(4\times9\). Exam tip: substitute the term number first, then perform multiplication before subtraction.

At (n=1) it gives (42), and at (n=2) it gives (37), so (a_n=47-5n). In exams, always match the first term in a decreasing sequence.

From (3n+2=38), we get (n=12). In exams, equate the formula to the given value when term number is asked.

Its rule is (a_n=3n+2), and (3n+2=32) gives (n=10). In exams, form the general term first to find the term number.

These terms are (2^2,3^2,4^2,5^2), so (a_n=(n+1)^2). In exams, recognize shifted square numbers.

Given (a_n=(n+1)^2), substitute n=6: (a_6=(6+1)^2=7^2=49). Option 36 results from squaring 6 without first adding 1, so it does not follow the given rule. Exam tip: substitute the term number into the rule, simplify the brackets first, and then apply the exponent.

The direct answer is option D: a_n=n(n+1). Substitute the position n into this product of two consecutive integers. At n=1, 1×2=2; at n=2, 2×3=6; at n=3, 3×4=12; at n=4, 4×5=20. Thus option D reproduces every listed term. Option A, n^2+1, gives 2,5,10,17, so the second term is wrong. Option B, 2n, gives 2,4,6,8 and has a constant difference, unlike the sequence. Option C, n^2+n+1, gives 3,7,13,21, so it is one more than the required value at each position. Option D is correct because the terms are products of consecutive numbers: 1×2, 2×3, 3×4, 4×5. The differences 4,6,8 also increase regularly, which supports this pattern. Memory cue: recognize n(n+1) whenever the terms look like consecutive-number products.

\(a_{n+1}-a_n=[3(n+1)-2]-(3n-2)=3\), so consecutive terms increase by a constant 3. Option C describes a multiplicative change, which does not apply here. Exam tip: find the common difference first.

Adding (1) to triangular numbers gives (2,4,7,11). In exams, think of triangular numbers when differences are (2,3,4).

The governing idea is direct substitution into an explicit formula. Since the question asks for a₅, replace n by 5 throughout the rule: a₅ = 5(5 + 1)/2 + 1 = 5 × 6/2 + 1. First calculate the product, 5 × 6 = 30; then divide by 2 to obtain 15; finally add the separate constant 1, giving 16. Therefore, option B is correct. A common error is to stop at 15 and forget the final +1, which leads to option A. Options C and D can result from adding too much or mishandling the fraction. The index n identifies the term, while the formula determines its value, so no pattern guess is needed.

Substituting \(n=1,2,3,4\) into \(a_n=n^2-1\) gives \(0,3,8,15\), respectively. Hence, \(a_n=n^2-1\) is the correct general term. For \(a_n=n^2+1\), the first term is \(2\), so it cannot represent the sequence. Exam tip: verify a proposed general term by checking at least \(n=1\) and \(n=2\).

To find the term number, set the explicit rule equal to the required value: n² − 1 = 35. Add 1 to both sides, giving n² = 36. Since a term index is a positive counting number, take n = √36 = 6, not the negative solution −6. Substituting back confirms the result: a₆ = 6² − 1 = 36 − 1 = 35. Therefore, option D is correct. Option A gives 4² − 1 = 15, option B gives 5² − 1 = 24, and option C gives 7² − 1 = 48. This illustrates that finding which term has a given value requires solving the explicit equation and then selecting a valid positive index.

Substituting \(n=1,2,3,4\) into \(a_n=2^n+1\) gives \(3,5,9,17\), respectively. Hence, the correct rule is \(a_n=2^n+1\). The closest distractor, \(2^n-1\), gives \(1\) as the first term, so it is incorrect. Exam tip: verify an explicit rule by substituting values of \(n\) for at least the first three terms.

Given (a_n=2^n+1), substitute n=5: (a_5=2^5+1=32+1=33). The value 31 would result from using (2^5-1), but the rule requires adding 1. Exam tip: substitute the term number first, then evaluate the exponent and remaining operations in order.