Explicit or general rule

Correct answer: C. 44

Explanation: 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.

Correct answer: D. 116

Explanation: 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.

Correct answer: B. 1

Explanation: 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.

Correct answer: B. 720

Explanation: 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.

Correct answer: C. 10th

Explanation: 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.

Correct answer: C. 7

Explanation: 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.

Correct answer: B. 503

Explanation: 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.

Correct answer: B. 2

Explanation: 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.

Correct answer: B. (a_n=5n-1)

Explanation: 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.

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

Explanation: 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.

Correct answer: A. (a_n=7n+5)

Explanation: 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).

Correct answer: C. 14

Explanation: 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.

Correct answer: B. (a_n=47-5n)

Explanation: 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.

Correct answer: D. (n=12)

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

Correct answer: C. (10)th term

Explanation: 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.

Correct answer: A. (a_n=(n+1)^2)

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

Correct answer: B. 49

Explanation: 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.

Correct answer: D. (a_n=n(n+1))

Explanation: 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.

Correct answer: A. Each next term increases by 3

Explanation: \(a_{n+1}-a_n=[3(n+1)-2]-(3n-2)=3\), 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.

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

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

Correct answer: B. 16

Explanation: 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.

Correct answer: A. \(a_n=n^2-1\)

Explanation: 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 \(n=2\).

Correct answer: D. n = 6

Explanation: 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.

Correct answer: C. \(a_n=2^n+1\)

Explanation: 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.

Correct answer: A. 33

Explanation: 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.