Explicit or general rule

At (n=1) it gives (188), and at (n=2) it gives (176), so (a_n=200-12n). In exams, check the first two terms of a decreasing sequence.

Substitute \(n=6\): \(a_6=\frac{6(5\times6-1)}{2}=\frac{6(30-1)}{2}=\frac{6\times29}{2}=87\). Hence, the correct answer is 87. A value such as 84 can result from an error while evaluating \(5n-1\). Exam tip: substitute the term number first, then simplify the bracket and perform multiplication and division step by step.

The governing concept is an explicit or general rule: one formula must produce the term for every positive integer n. Test option A by substituting n = 1, 2, 3 and 4: a₁ = 1(5−1)/2 = 2, a₂ = 2(10−1)/2 = 9, a₃ = 3(15−1)/2 = 21, and a₄ = 4(20−1)/2 = 38. Therefore option A reproduces all given terms. Option B gives 3, 7, 12, 18, while option C gives 2, 9, 16, 23 and option D gives 2, 8, 18, 32, so those distractors fail the direct substitution check.

Given \(a_n=(n+1)^3-n\), substitute \(n=4\): \(a_4=(4+1)^3-4=5^3-4=125-4=121\). Therefore, 121 is the correct option. 123 would result from an incorrect subtraction. Exam tip: substitute the term number first, evaluate the exponent, and then subtract.

Substituting \(n=1,2,3,4\) into \(a_n=(n+1)^3-n\) gives \(7,25,61,121\), respectively. Therefore, option A is correct. Option B gives the first term as \(7\), but for \(n=2\) it gives \(14\), not \(25\). Exam tip: verify a proposed general term using at least the first three terms.

Substituting \(n=4\), \(a_4=3\cdot2^4+2(4)^2=3\cdot16+2\cdot16=48+32=80\). Therefore, 80 is correct. An answer such as 76 can result from incorrectly evaluating either the exponential or squared term. Exam tip: calculate \(2^4\) and \(4^2\) separately before multiplying by their coefficients.

The correct rule is \(a_n=3\cdot2^n+2n^2\). Checking it: for \(n=1\), \(3\cdot2+2=8\); for \(n=2\), \(3\cdot4+8=20\); and for \(n=3\), \(3\cdot8+18=42\). Option C gives the first term \(8\), but for \(n=2\) it gives \(16\), not \(20\). Exam tip: test an explicit rule using at least the first three terms.

Given \(a_n=7n^2-4n+1\), substitute \(n=5\): \(a_5=7(5)^2-4(5)+1=7\times25-20+1=175-20+1=156\). Hence, 156 is correct. The value 154 may result from incorrectly omitting the final \(+1\). Exam tip: substitute the term number first, then evaluate powers and multiplication carefully.

(7n^2-4n+1) gives (4,21,52,97). In exams, identify a quadratic rule by equal second differences.

The consecutive differences are \(19,29,39\), whose second differences are \(10\). Therefore, the sequence has a quadratic general term of the form \(a_n=5n^2+bn+c\), since its second difference is \(2\times5=10\). Substituting \(n=1\) and \(n=2\) gives \(b=4\) and \(c=-1\), so \(a_n=5n^2+4n-1\) is correct. The close distractor \(a_n=5n^2+2n+1\) gives the first term correctly but gives \(25\), not \(27\), for the second term. Exam tip: verify a proposed general term using at least the first two or three terms.

Using \(a_n=\frac{n(3n+5)}{2}\), substitute \(n=8\): \(a_8=\frac{8(3\times8+5)}{2}=\frac{8(29)}{2}=4\times29=116\). Hence, option C is correct. A value such as \(112\) can result from an error while evaluating the expression inside the bracket. Exam tip: substitute the value of \(n\) first, then simplify brackets and multiplication/division step by step.

(2\cdot5^{n-1}+n^2) gives the given terms. In exams, check both the power and square in rapid growth.

From the given terms, (p+q=5) and (2p+q=7), so (p=2), (q=3), and (a_4=45). When coefficients are unknown, first form equations using small terms.

The governing concept is the explicit or general rule for an arithmetic progression. The first term is a₁ = 8, and the common difference is d = 13 − 8 = 5. Use aₙ = a₁ + (n − 1)d. Substitution gives aₙ = 8 + (n − 1)5 = 8 + 5n − 5 = 5n + 3, so option A is correct. Checking confirms it: for n = 1, the rule gives 8; for n = 2, it gives 13; and for n = 3, it gives 18. Option B gives 8 as the first term but then has difference 8, while option C gives 2 for n = 1 and option D has common difference 1.

The governing concept is evaluating an explicit sequence rule at the correct index. In the usual indexing of a sequence, the first term is obtained by setting n = 1, not n = 0. Therefore a₁ = 30 − 4(1) = 30 − 4 = 26, so option B is correct. The rule also has common difference −4, since aₙ₊₁ − aₙ = [30 − 4(n + 1)] − [30 − 4n] = −4, confirming that it describes an arithmetic progression. Option A is the constant term in the formula, not the first sequence value. Option C would result from incorrectly using n = 1.5, while option D does not follow from the stated rule.

The governing idea is an explicit or general rule: substitute the term numbers into the formula. For the first term, put n = 1: a_1 = 3(1) + 4 = 7. For the second term, put n = 2: a_2 = 3(2) + 4 = 10. For the third term, put n = 3: a_3 = 3(3) + 4 = 13. Thus the first three terms are 7, 10, 13, so option B is correct. Option A begins with 4, which would result from using n = 0; options C and D come from incorrect substitution or arithmetic.

For a general term of an arithmetic progression, use a_n = a + (n − 1)d. The first term is a = 10 and the common difference is d = 15 − 10 = 5. Therefore a_n = 10 + (n − 1)5 = 10 + 5n − 5 = 5n + 5. Option A is correct. Checking confirms it: for n = 1, the rule gives 10; for n = 2, it gives 15; and for n = 3, it gives 20. Option B gives 10 as the first term but then increases by 10, option C begins at zero, and option D increases by only 1, so none of them describes the given progression.

The governing concept is the explicit or general term of an arithmetic progression, a_n = a + (n − 1)d. The first term is a = 4, and the common difference is d = 11 − 4 = 7. Substitution gives a_n = 4 + (n − 1)7 = 4 + 7n − 7 = 7n − 3. Therefore option A is correct. A quick check is useful: for n = 1, option A gives 7(1) − 3 = 4, and for n = 2 it gives 14 − 3 = 11, matching the sequence. Option B gives 11 for the first term, option C gives 11, and option D gives 10; hence they fail the initial-term test even before later terms are considered.

The governing concept is an explicit or general rule for a sequence. The notation a_n=4n+3 gives the value of the nth term directly, so substitute the first four positive integer values of n. For n=1, a_1=4(1)+3=7; for n=2, a_2=4(2)+3=11; for n=3, a_3=4(3)+3=15; and for n=4, a_4=4(4)+3=19. Hence option A, (7, 11, 15, 19), is correct. Option C begins with the value obtained from n=0, not the first term under the usual indexing n=1. Options B and D use an incorrect starting value or constant.

The governing concept is an explicit or general rule for a sequence: the formula gives the value directly from the term number n. To find the position of a specified value, set the rule equal to that value. Thus, 9n + 4 = 85. Subtracting 4 gives 9n = 81, and dividing by 9 gives n = 9. Therefore, 85 occurs at the ninth term, so option C is correct. Verification is immediate: a₉ = 9(9) + 4 = 81 + 4 = 85. The other choices produce values 67, 76, and 94 when substituted into the rule.