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 evaluating an explicit rule that contains an alternating sign term. For n = 10, the exponent is even, so (−1)^10 = 1; hence a_10 = 5(10) + 1 = 51. For n = 9, the exponent is odd, so (−1)^9 = −1; hence a_9 = 5(9) − 1 = 44. Their difference is a_10 − a_9 = 51 − 44 = 7, so option C is correct. A common error is to treat both powers of −1 as positive, which would give 5, or to mishandle the subtraction of the negative term. The alternating component must be evaluated separately for each index.

The governing concept is substitution in an explicit, or general, term formula. The expression a_n = 4n^2 + 2n - 3 gives the value of any term directly when its position n is known. For the eleventh term, substitute n = 11: a_11 = 4(11^2) + 2(11) - 3. Since 11^2 = 121, this becomes 4(121) + 22 - 3 = 484 + 22 - 3 = 503. Therefore option B is correct. The other choices can result from squaring or adding incorrectly, or from using a nearby term number.

The governing ideas are direct evaluation of an explicit sequence formula and careful handling of alternating signs. For n = 8, the sign factor is (-1)^9 = -1, so a_8 = -(2×8+3) = -19. For n = 9, the sign factor is (-1)^10 = +1, so a_9 = 2×9+3 = 21. Adding the two values gives a_8 + a_9 = -19 + 21 = 2. Hence option B is correct. Option A would incorrectly treat the magnitudes as equal, while options C and D arise from ignoring or mishandling the alternating sign.

The governing concept is solving for the term number in an explicit formula. We need n^2 + 6n + 5 = 252, so n^2 + 6n - 247 = 0. Factoring gives (n + 13)(n - 19) = 0, which would suggest n = 19, not one of the listed options; therefore the supplied key and options are inconsistent with the stated formula. Direct checking confirms a_13 = 169 + 78 + 5 = 252. Thus, for the intended school-MCQ answer, option C is correct, but the formula itself should be corrected to n^2 + 4n + 36 if 13 is to be its exact solution, or the target/options should be revised. This item requires revision rather than a pass.

The governing concept is an explicit or general rule, which expresses a term directly in terms of its position n rather than using the preceding term. The sequence 3, 6, 9, 12 consists of successive multiples of 3. Testing aₙ = 3n gives a₁ = 3(1) = 3, a₂ = 3(2) = 6, a₃ = 3(3) = 9 and a₄ = 3(4) = 12, so it matches every displayed term. Therefore option C is correct. The rule n + 3 gives 4 as its first term, 2n gives 2 as its first term, and 3n + 1 gives 4 as its first term. Since each of these already fails at n = 1, none can represent the sequence.

The governing concept is an explicit rule, where the nth term is obtained directly from n. The displayed sequence is made of consecutive multiples of 5: 5×1, 5×2, 5×3 and 5×4. Thus the natural rule is aₙ = 5n. Checking it gives a₁ = 5, a₂ = 10, a₃ = 15 and a₄ = 20, exactly as required. Therefore option A is correct. The rule n + 5 produces 6 as the first term, 4n produces 4 as the first term, and 5n − 1 produces 4 as the first term; each fails immediately at n = 1. The constant difference is 5, but that fact must still be combined with the correct first-term position to obtain 5n.

This is the sequence of even numbers, so (a_n=2n). In exams, remember (2n) for even numbers.

This is the sequence of odd numbers, so (a_n=2n-1). In exams, check the rule by putting (n=1).

For the first three terms, substitute \(n=1,2,3\). This gives \(a_1=1^2=1\), \(a_2=2^2=4\), and \(a_3=3^2=9\). Therefore, the sequence begins as \((1,4,9)\). Option A lists the values of \(n\), not their squares. Exam tip: Substitute \(n=1,2,3\) into the general term to find the initial terms quickly.

The governing concept is the explicit formula for an arithmetic sequence. The terms increase by a constant common difference, d = 7 − 4 = 3. For an arithmetic sequence, a_n = a_1 + (n − 1)d. Substituting a_1 = 4 and d = 3 gives a_n = 4 + 3(n − 1) = 4 + 3n − 3 = 3n + 1. Checking n = 1 gives 4, n = 2 gives 7, and n = 3 gives 10, so option A matches every listed term. Option B gives 4, 8, 12, while option C starts with 2 and option D gives 4, 5, 6; therefore they do not describe the sequence.

For \(a_n=3n-2\), \(a_{n+1}-a_n=3\), which is constant; hence it is an arithmetic progression. In \(n^2+1\), the differences vary. Exam tip: check the first difference.

This question tests recognition of an explicit rule. Every displayed term is obtained by multiplying its position number by 10: the first term is 10 × 1, the second is 10 × 2, the third is 10 × 3, and the fourth is 10 × 4. Therefore the general term is a_n = 10n, so option C is correct. It can also be verified using the arithmetic-sequence formula: a_1 = 10 and d = 10, hence a_n = 10 + (n − 1)10 = 10n. Option A gives 11 as its second term, option B gives 5, 10, 15, …, and option D gives 9 for n = 1. Thus none of those alternatives reproduces all the listed terms.

In \(a_n=2n+1\), any term is obtained directly by substituting its position \(n\), so it is explicit. Options A and D depend on a previous term and are recursive rules. Exam tip: an explicit rule usually expresses \(a_n\) directly in terms of \(n\).

The first term is (6) and the difference is (5), so (a_n=5n+1). In exams, check the formula on the first two terms.

For the first term, putting (n=1) gives (a_1=10). In exams, always put (n=1) for the first term.

This is an arithmetic progression with first term \(a=9\) and common difference \(d=9\). Thus, \(a_n=a+(n-1)d=9+(n-1)\times9=9n\). Therefore, the correct rule is \(a_n=9n\). The rule \(a_n=9n+1\) gives 10 as the first term, so it is incorrect. Exam tip: substitute \(n=1\) to check whether a proposed rule gives the first term correctly.