Explicit or general rule

The governing concept is testing an explicit rule against the sequence, while the first differences help identify its form. The differences are 7, 11, and 15; their second differences are 4 and 4, so a quadratic rule is plausible. Test option A: for n = 1, 2(1)^2 + 1 + 2 = 5; for n = 2, 2(2)^2 + 2 + 2 = 12; for n = 3, 18 + 3 + 2 = 23; and for n = 4, 32 + 4 + 2 = 38. Every listed term is reproduced, so option A is correct. Option B gives 5, 12, 21, 32, while C gives 5, 11, 21, 35; D already gives 5, 11, 17, 23. Thus the distractors fail on later terms.

The governing concept is evaluating an explicit formula at successive positive integer indices. Compute each term carefully, remembering that 2^n is an exponential part and 3n is a separate linear part. For n = 1, a_1 = 2^1 + 3(1) = 2 + 3 = 5. For n = 2, a_2 = 2^2 + 3(2) = 4 + 6 = 10. For n = 3, a_3 = 8 + 9 = 17. For n = 4, a_4 = 16 + 12 = 28. Hence the first four terms are (5, 10, 17, 28), so option A is correct. Option B effectively undercounts the linear contribution, while C and D introduce incorrect additions at later indices. The exponent applies only to 2, not to the entire expression.

The magnitude is (6,11,16,21,\ldots) and signs start positive and alternate, so (a_n=(-1)^{n+1}(5n+1)). The sign of the first term decides the power.

The change over six gaps is (-36), so (d=-6), hence (a_n=31-6n). From two given terms first find the common difference.

Given \(n^2+5n=126\), we get \(n^2+5n-126=0\). Factoring gives \((n-9)(n+14)=0\), so \(n=9\) or \(n=-14\). Since a term position \(n\) in a sequence is positive, \(n=9\) is correct. For example, \(n=8\) gives 104, not 126. Exam tip: bring all terms to one side and factor the quadratic expression.

The first differences are \(4,6,8\), and the second differences are constant: \(2,2\). Hence, the general term should be quadratic. Substituting \(n=1,2,3,4\) in \(a_n=n^2+n+2\) gives \(4,8,14,22\), respectively. Although \(n^2+3\) gives the first term as \(4\), it gives \(7\) as the second term, so it is incorrect. Exam tip: when second differences are constant, test a quadratic rule using the first few terms.

(a_4=\frac{4(4)+1}{3(4)-1}=\frac{17}{11}). In a fractional rule substitute (n) in both numerator and denominator.

This is (6^3,7^3,8^3,9^3,\ldots), so (a_n=(n+5)^3). In cube sequences observe the order of base numbers.

In option A, substituting \(n=1,2,3,4\) gives \(14,31,54,83\), respectively. Therefore, the correct rule is \(a_n=3n^2+8n+3\). For example, when \(n=3\), \(a_3=3(3)^2+8(3)+3=54\). Option B also gives 14 as its first term, but it gives 29, not 31, when \(n=2\). Exam tip: verify at least the first two or three terms before selecting a general rule.

In a linear rule the coefficient of (n) is (-5), so the common difference is (-5). Do not miss the sign while reading the coefficient.

The governing concept is evaluating a formula at two different indices and then adding the resulting terms. First calculate a_4: a_4 = 6(4^2) - 1 = 6(16) - 1 = 96 - 1 = 95. Next calculate a_2: a_2 = 6(2^2) - 1 = 6(4) - 1 = 24 - 1 = 23. Therefore, a_4 + a_2 = 95 + 23 = 118. Option A is correct. A likely error is to omit the -1 twice or to add the indices before applying the formula; those mistakes can produce distractor values such as 120 or other nearby numbers. Each term must be evaluated separately because the expression asks for the sum of two sequence terms, not the value of the rule at n = 6.

The general term is (a_n=79-9n), and (79-9n=-2) gives (n=9). Even in decreasing sequences the position is natural.

Substitute n=1, 2, and 3 successively. This gives a_1=4(1)^2-3(1)+8=9, a_2=4(2)^2-3(2)+8=18, and a_3=4(3)^2-3(3)+8=35. Hence, the correct sequence is (9,18,35). Option (9,20,39) has the correct first term, but its calculations for n=2 and n=3 are incorrect. Exam tip: substitute each value of n separately, carrying out squaring and multiplication first.

The numerator is (n^2+5) and the denominator is (n^2+4), so (a_n=\frac{n^2+5}{n^2+4}). Identify square patterns in fractional sequences.

The rule of a sequence can be found by observing how the numerator and denominator depend on n. The numerators are 3, 6, 9, and 12, which are exactly 3n for n=1, 2, 3, and 4. The denominators are 7, 12, 17, and 22. They increase by 5, and the expression 5n+2 gives these values. Thus the general term is \(a_n=\frac{3n}{5n+2}\), which is option A.

Substitution verifies the answer: at n=1, \(\frac{3}{7}\) is obtained; at n=2, \(\frac{6}{12}\); at n=3, \(\frac{9}{17}\); and at n=4, \(\frac{12}{22}\). Option C uses the wrong denominator pattern, and options B and D do not produce the numerators 3n. Both parts of the fraction must match the sequence.

The governing concept is evaluating two terms from an explicit linear rule and then finding their difference. For n = 9, a_9 = 13(9) - 8 = 117 - 8 = 109. For n = 4, a_4 = 13(4) - 8 = 52 - 8 = 44. Therefore, a_9 - a_4 = 109 - 44 = 65, so option B is correct. There is also a useful shortcut: because the coefficient of n is 13, increasing the index from 4 to 9 by 5 increases the term by 5 × 13 = 65; the constant -8 cancels in the subtraction. Option A corresponds to only four index steps, while C and D overcount the change. Both direct calculation and the shortcut confirm 65.

The magnitude is (7n) and signs start positive and alternate, so (a_n=(-1)^{n+1}7n). Choose the power of ((-1)) by checking the first term sign.

The governing concept is recognizing and verifying a quadratic explicit rule. The first differences are 13, 23, and 33; their second differences are both 10, which is consistent with a quadratic expression whose n^2 coefficient is 5. Test option A directly: at n = 1, 5 - 2 + 6 = 9; at n = 2, 20 - 4 + 6 = 22; at n = 3, 45 - 6 + 6 = 45; and at n = 4, 80 - 8 + 6 = 78. It reproduces every given term, so option A is correct. Option B gives 9, 20, 41, 72; C gives 9, 24, 47, 78; and D gives 9, 25, 47, 75. Matching only the first term is insufficient, so later substitution distinguishes the correct rule.

Because the fifth term is known, substitute n = 5 into the explicit formula. We obtain a₅ = 5² + 5q + 4 = 25 + 5q + 4 = 29 + 5q. Equating this to 54 gives 29 + 5q = 54, so 5q = 25 and q = 5. Hence option C is correct. The other choices arise from an incorrect substitution of 5 or from mishandling the constant term.

(a_5=2) and (a_6=-2), so the first negative term is the (6)th. Do not count zero or positive terms as negative.