Explicit or general rule

The governing concept is generating sequence terms from an explicit rule by using n = 1, 2, 3, and 4. For n = 1, a_1 = 1 + 6 + 5 = 12. For n = 2, a_2 = 4 + 12 + 5 = 21. For n = 3, a_3 = 9 + 18 + 5 = 32. For n = 4, a_4 = 16 + 24 + 5 = 45. Hence the first four terms are 12, 21, 32, 45, so A is correct. Starting with n = 0 would produce a different list, but sequence notation normally begins with n = 1 here; the other options reflect arithmetic or indexing errors.

The numerator is (2n+1) and the denominator is (3n+2), so (a_n=\frac{2n+1}{3n+2}). In fractions observe numerator and denominator patterns separately.

The governing concept is direct substitution into an explicit product-form sequence rule. Use n = 1, 2, 3, and 4. We obtain a_1 = 1(1 + 3) = 4, a_2 = 2(2 + 3) = 10, a_3 = 3(3 + 3) = 18, and a_4 = 4(4 + 3) = 28. Therefore option B gives the first four terms. Option A begins with the value for n = 0, while option C corresponds to n(n + 1), not n(n + 3). Option D does not follow the stated multiplication rule. Keeping the two factors visible reduces errors when substituting each index.

For the fifth term, substitute \(n=5\): \(a_5=4\cdot3^{5-1}=4\cdot3^4=4\cdot81=324\). Therefore, 324 is correct. The value 216 can result from incorrectly using \(3^3\) and miscounting the exponent \(n-1\). Exam tip: after substituting the term number, evaluate \(n-1\) before calculating the power.

This is (5^2,6^2,7^2,8^2,\ldots), so (a_n=(n+4)^2). In square sequences relate the base number to (n).

(a_6=23) and (b_4=24), so the sum is (47). With two rules the term numbers may differ, so use them carefully.

Its rule is (a_n=3n^2+4n+4), so the (10)th term is (344), not any listed option. In exams also check the consistency of options.

Substituting \(n=1,2,3,4\) into \(a_n=n^2+6n+1\) gives \(8,17,28,41\), respectively. Therefore, option A is correct. In option B, the term for \(n=2\) is \(18\), not the given second term \(17\). Exam tip: Verify a proposed sequence rule by substituting at least the first two or three values of \(n\).

The general term is (a_n=8n+5), and (8n+5=93) gives (n=11). If (n) is a natural number, the given term belongs to the sequence.

Substitute \(n=3\) in the given rule: \(a_3=6^3-3(3)=216-9=207\). Therefore, 207 is the correct option. The value 213 would result from not subtracting the \(3n\) term, so it is incorrect. Exam tip: For a term of a sequence, substitute the value of \(n\) everywhere first, then evaluate powers and the remaining operations.

The numerator is (3n+2) and the denominator is (4n+2), so (a_n=\frac{3n+2}{4n+2}). In a fractional sequence form separate rules for both parts.

The increase over five gaps is (35), so (d=7), then (a_n=7n-5). From two given terms first find the common difference.

Substituting n=1, 2, and 3 into the rule gives a_1=3(1)^2-2(1)+6=7, a_2=3(2)^2-2(2)+6=14, and a_3=3(3)^2-2(3)+6=27. Hence, the correct sequence is (7, 14, 27). Option D has the correct first term, but for n=2 the value is 14, not 16. Exam tip: substitute each value of n carefully into the entire expression.

The governing concept is substitution into an explicit sequence rule. The notation a_8 means that n must be replaced by 8 in a_n = pn + 7. Thus, 8p + 7 = 71. Subtracting 7 from both sides gives 8p = 64, and dividing by 8 gives p = 8. A quick check confirms the result: a_8 = 8(8) + 7 = 64 + 7 = 71. Option A would give 55, option B would give 63, and option D would give 79, so none of them satisfies the given eighth term. Therefore, option C is the only correct answer. The important point is to use the index 8, not the first term or the coefficient itself, when applying the rule.

The governing concept is the explicit formula for an arithmetic sequence: a_n = a_1 + (n - 1)d. Here the first term is a_1 = 17, and each term is 9 less than the preceding one, so the common difference is d = -9. Substitution gives a_n = 17 + (n - 1)(-9) = 17 - 9n + 9 = 26 - 9n. Checking n = 1 gives 26 - 9 = 17, and n = 2 gives 26 - 18 = 8, which is indeed 9 less. Option A gives 8 at n = 1, because it forgets the n - 1 adjustment. Option C increases rather than decreases, and D also has the wrong direction. Thus option B is correct.

This is three times the triangular numbers, so (a_n=\frac{3n(n+1)}{2}). Recognize the pattern from additions (6,9,12).

For option A, substituting \(n=1,2,3,4\) gives \(12,25,42,63\), respectively. Also, the first differences are \(13,17,21\), so the second differences are constant at \(4\); this supports a quadratic rule. Option C matches the first two terms but gives \(40\), not \(42\), when \(n=3\). Exam tip: test a proposed rule with at least three terms, not just the first term.

From (\frac{3n}{n+4}=\frac{9}{7}), (21n=9n+36), so (n=3). None of the given options is correct.

Given \(a_n=4n^2+3\), \(a_3=4(3)^2+3=39\) and \(a_5=4(5)^2+3=103\). Therefore, \(a_3+a_5=39+103=142\). A value such as 140 can result from an error while squaring or adding. Exam tip: substitute each required value of \(n\) separately into the general term before finding their sum.

This is (4^2,8^2,12^2,16^2,\ldots), so (a_n=(4n)^2=16n^2). In square sequences with equal base gaps, find the base rule.