Explicit or general rule

(4n^2+5n-6) gives (3,20,45,78). In exams, choose a quadratic rule when second differences are constant.

Substitute n=4: \(a_4=4^3+3(4^2)-2=64+48-2=110\). Therefore, 110 is the correct option. A nearby distractor such as 108 results from an incorrect final calculation; \(3\times4^2\) must be evaluated as 48. Exam tip: calculate powers first, then perform multiplication and addition or subtraction.

For \(a_n=n^3+3n^2-2\), substituting \(n=1,2,3,4\) gives \(2,18,52,110\), respectively. Hence, it is the correct general term. The close distractor \(2n^3\) gives the first term as 2, but gives 16 rather than 18 when \(n=2\). In an exam, verify a proposed general term using at least the first two or three terms.

Substitute \(n=3\) in the given rule: \(a_3=2(3)^3+(3)^2+3=2\times27+9+3=54+9+3=66\). Therefore, 66 is the correct option. Getting 64 indicates an error in evaluating the cubic term \(2(3)^3\). Exam tip: substitute the value in every term first, then evaluate powers before adding.

The correct rule is \(a_n=2n^3+n^2+n\). Substituting \(n=1,2,3,4\) gives \(4,22,66,148\), respectively. The closest distractor, \(4n^2\), gives 4 when \(n=1\), but it gives 16 rather than 22 when \(n=2\). Exam tip: verify a proposed general term using at least the first three terms.

Substitute \(n=4\): \(a_4=\frac{3(4)^2+5(4)}{4}=\frac{3\times16+20}{4}=\frac{68}{4}=17\). Therefore, 17 is correct. Getting 16 may result from incorrectly evaluating \(3\times4^2\) or making an error in the numerator. Exam tip: substitute the value of \(n\) first, evaluate the power, and then simplify the numerator and denominator.

The governing idea is verification of an explicit general term by substituting successive values of n. For option A, n = 1 gives (3 + 5)/4 = 2. For n = 2, (3·4 + 10)/4 = 22/4 = 11/2. For n = 3, (3·9 + 15)/4 = 42/4 = 21/2. For n = 4, (3·16 + 20)/4 = 68/4 = 17. Thus A reproduces all four terms exactly. Option B gives 2, 5 and 9, option C gives 2, 5 and 8, and option D gives 3/2 for the first term. Consequently, fractional terms must be checked carefully rather than rounded or treated as whole numbers.

The governing concept is evaluating an explicit formula and then simplifying a ratio. First calculate a₄: 6(4²) − 4 + 4 = 6·16 = 96. Next calculate a₂: 6(2²) − 2 + 4 = 24 − 2 + 4 = 26. Hence a₄ : a₂ = 96 : 26. Both numbers are divisible by 2, so the ratio in simplest form is 48 : 13, making option B correct. Option A is the correct unsimplified ratio, but the question asks for the ratio and the simplified option is preferred. Option C reverses the order, while option D uses an incorrect value for a₄.

The governing concept is identifying a quadratic explicit rule. The first differences are 17, 29 and 41; their second differences are 12 and 12, so a quadratic expression is appropriate. Test option A directly: a₁ = 6(1)² − 1 + 4 = 9; a₂ = 6(4) − 2 + 4 = 26; a₃ = 6(9) − 3 + 4 = 55; and a₄ = 6(16) − 4 + 4 = 96. It matches every listed term, so A is correct. Option B is linear and gives 18 at n = 2, option C gives 26 at n = 2 but 43 at n = 3, and option D gives 9 at n = 1 but 28 at n = 2. Constant second differences support the quadratic rule.

Substitute 3 for n in the general term: \(a_3=5^3+3-4=125+3-4=124\). Therefore, the correct answer is 124. A value such as 122 can result from an arithmetic error in the addition or subtraction. Exam tip: evaluate the power first, then add or subtract the remaining terms.

The governing concept is an explicit rule containing exponential growth. Substitute the position n into option A: for n = 1, 5¹ + 1 − 4 = 2; for n = 2, 5² + 2 − 4 = 23; for n = 3, 5³ + 3 − 4 = 124; and for n = 4, 5⁴ + 4 − 4 = 625. Thus the formula exactly generates the four given terms, so A is correct. Option B is linear and gives 7 for the first term, option C gives 7 for the first term, and option D gives −3 for n = 1 and 28 for n = 2. The small adjustment n − 4 must not be omitted when checking the power pattern.

The first five terms are (3,16,29,42,55), and the average is (29). In exams, divide the sum by the number of terms.

Its rule is (a_n=13n-10), and (13n-10=159) gives (n=13). In exams, equate the given term to the general term.

Substitute n=3: a_3=3^4-3^2=81-9=72. Therefore, 72 is the correct option. A value such as 70 can result from an error while evaluating the powers or subtracting. Exam tip: calculate each power separately before subtracting.

For \(a_n=n^4-n^2\), we get \(a_1=1-1=0\), \(a_2=16-4=12\), \(a_3=81-9=72\), and \(a_4=256-16=240\). Hence, the correct general term is \(a_n=n^4-n^2\). The close distractor \(2n^4\) gives \(a_1=2\), which does not match the first term 0. Exam tip: substitute \(n=1,2,3\) to verify a proposed general term against the initial terms.

Given \(a_n=4^n+n-3\), substitute \(n=4\): \(a_4=4^4+4-3=256+4-3=257\). Therefore, 257 is correct. A value such as 255 can result from an error while evaluating the \(n-3\) part. Exam tip: after substitution in a general term, evaluate the exponent first and then perform addition and subtraction carefully.

Substituting \(n=1,2,3,4\) into \(a_n=4^n+n-3\) gives \(2,15,64,257\), respectively. Although \(a_n=n^4+1\) gives the first term as 2, its second term is \(17\), not 15. Exam tip: verify an explicit rule by testing it for at least the first three terms.

Given \(a_n=2n^2+9n+1\). Substituting \(n=5\), \(a_5=2(5)^2+9(5)+1=2\times25+45+1=96\). Hence, \(96\) is the correct option. A value such as \(94\) can result from an error while evaluating the squared term \(2\times25\). Exam tip: substitute the value of \(n\) first, then evaluate the exponent carefully.

The governing concept is finding an explicit quadratic rule from the sequence. The first differences are 15, 19 and 23, which increase by 4, so the second difference is constant and a quadratic formula is expected. Check option A: for n = 1, 2 + 9 + 1 = 12; for n = 2, 8 + 18 + 1 = 27; for n = 3, 18 + 27 + 1 = 46; and for n = 4, 32 + 36 + 1 = 69. Therefore A matches all four terms. Option B and C are linear and cannot reflect the changing first differences. Option D gives 11 for n = 1 and 28 for n = 2, so it fails immediately.

The given rule is \(a_n=200-12n\). Substituting \(n=9\), we get \(a_9=200-12\times9=200-108=92\). Therefore, 92 is correct. A value such as 88 results from an arithmetic error after substitution. Exam tip: substitute the term number first, then perform multiplication before subtraction.