Explicit or general rule

Correct answer: A. \(a_n=n^2+4n+1\)

Explanation: For option A, substituting \(n=1,2,3,4\) gives \(6,13,22,33\), respectively. Hence, the correct general term is \(a_n=n^2+4n+1\). The closest distractor, option D, gives the first two terms as \(6,13\), but for \(n=3\) it gives \(20\), not \(22\). Exam tip: test a proposed general term using at least the first three values of \(n\).

Correct answer: C. 48

Explanation: Given \(a_n=\frac{n(n+4)}{2}\). For \(a_8\), substitute \(n=8\): \(a_8=\frac{8(8+4)}{2}=\frac{8\times12}{2}=48\). Hence, 48 is the correct option. Values such as 46 or 50 can result from an arithmetic error, but \(8+4=12\). Exam tip: Substitute the required value of \(n\) first, then simplify step by step.

Correct answer: A. (a_n=\frac{n(n+4)}{2})

Explanation: Using (\frac{n(n+4)}{2}) gives the given terms. In exams check fractional terms using small (n) values.

Correct answer: B. 26

Explanation: Given \(a_n=2^n+3n-2\). Substituting \(n=4\), we get \(a_4=2^4+3(4)-2=16+12-2=26\). Therefore, option B is correct. Choosing \(28\) usually results from an error in addition or subtraction. Exam tip: evaluate the exponent first, then add or subtract the remaining terms in order.

Correct answer: A. (a_n=2^n+3n-2)

Explanation: (2^n+3n-2) gives (3,8,17,36). In exams also check the extra linear part in a power-based rule.

Correct answer: B. 39 : 8

Explanation: The governing concept is substitution into an explicit sequence rule followed by reduction of a ratio. For n = 4, calculate a₄ = 6(4²) − 5(4) + 2 = 96 − 20 + 2 = 78. For n = 2, calculate a₂ = 6(2²) − 5(2) + 2 = 24 − 10 + 2 = 16. Hence a₄ : a₂ = 78 : 16. The common factor of 78 and 16 is 2, so divide both parts by 2 to obtain 39 : 8. Therefore option B is correct. Option A contains the correct unsimplified values but is not in simplest form. Options C and D result from incorrect substitution or arithmetic and do not represent the given formula.

Correct answer: A. aₙ = 6n² − 5n + 2

Explanation: The governing concept is finding an explicit general term from the observed pattern of a sequence. The first differences are 16−3=13, 41−16=25, and 78−41=37. Their second differences are 25−13=12 and 37−25=12, so a quadratic rule is appropriate. Test option A at each displayed position: when n=1, 6(1)²−5(1)+2=3; when n=2, 24−10+2=16; when n=3, 54−15+2=41; and when n=4, 96−20+2=78. It reproduces every term, so A is correct. Option C is linear and cannot have changing first differences. Options B and D fail when checked against one or more given terms.

Correct answer: C. 101

Explanation: Given \(a_n=3\cdot2^n+n\). Substituting \(n=5\), \(a_5=3\cdot2^5+5=3\cdot32+5=96+5=101\). Hence, 101 is the correct option. The value 96 comes from calculating only \(3\cdot2^5\) and incorrectly omitting the final \(+5\). Exam tip: Substitute the term number in every part of the formula before simplifying.

Correct answer: A. (a_n=3\cdot2^n+n)

Explanation: (3\cdot2^n+n) gives (7,14,27,52). In exams check both the power and extra term in rapid growth.

Correct answer: B. (29.5)

Explanation: The first six terms are (2,13,24,35,46,57) and the average is (29.5). In exams divide the sum by the number of terms.

Correct answer: B. (13)th term

Explanation: The rule is (a_n=11n-9) and (11n-9=134) gives (n=13). In exams equate the given term to the general term.

Correct answer: C. 92

Explanation: Given (a_n=n^3+2n^2-n), substitute n=4: (a_4=4^3+2(4^2)-4=64+32-4=92). Therefore, 92 is the correct option. A value such as 94 can result from an error while subtracting the final 4. Exam tip: Substitute the value of n carefully into every term of the rule.

Correct answer: A. \(a_n=\frac{9n^3-29n^2+48n-24}{2}\)

Explanation: The first differences are \(12,37,89\), and the second differences are \(25,52\). The third difference is constant at \(27\), so an appropriate rule is a cubic polynomial. Substituting \(n=1,2,3,4\) in \(a_n=\frac{9n^3-29n^2+48n-24}{2}\) gives \(2,14,51,140\), respectively. Option B gives \(42\) when \(n=3\), not \(51\). Exam tip: verify a cubic-type rule by substituting at least the first three or four values of \(n\).

Correct answer: C. (56)

Explanation: (a_2=74) and (a_9=18) so the difference is (56). In exams find both terms carefully in a decreasing formula.

Correct answer: B. (a_n=90-8n)

Explanation: At (n=1) it gives (82) and at (n=2) it gives (74) so (a_n=90-8n). In exams check the first two terms of a decreasing sequence.

Correct answer: A. \(a_2=27\) and \(a_3=64\)

Explanation: For the second term, substitute \(n=2\): \(a_2=7(2)^2+2(2)-5=28+4-5=27\). Similarly, for \(n=3\): \(a_3=7(3)^2+2(3)-5=63+6-5=64\). Hence, option A is correct. The other options result from an error in evaluating the \(n^2\) term or in addition/subtraction. Exam tip: substitute the value of \(n\) separately for each required term and check the arithmetic.

Correct answer: A. (a_n=7n^2+2n-5)

Explanation: The direct answer is option A, \(a_n=7n^2+2n-5\). Substitute each position carefully. For \(n=1\), \(7(1)^2+2(1)-5=4\). For \(n=2\), \(7(4)+4-5=27\). For \(n=3\), \(7(9)+6-5=64\). For \(n=4\), \(7(16)+8-5=115\). Thus the formula produces every listed term. The first differences are 23, 37, and 51; their differences are 14 and 14, so a quadratic rule is reasonable. Option B, \(4n^2\), gives 4 but then 16 rather than 27. Option C gives 4 at \(n=1\), but 31 at \(n=2\), not 27. Option D gives a linear sequence beginning 4, 27, 50, 73, so it fails after two terms. Memory cue: constant second differences suggest a quadratic rule, but always substitute several values.

Correct answer: C. 65

Explanation: Given \(a_n=4^n+2n-5\), substitute \(n=3\): \(a_3=4^3+2(3)-5=64+6-5=65\). Therefore, 65 is correct. A value such as 63 can result from incorrectly evaluating or adding the \(2n\) term. Exam tip: substitute the term number into every part of the rule before simplifying.

Correct answer: A. (a_n=4^n+2n-5)

Explanation: The direct answer is option A, \(a_n=4^n+2n-5\). The exponent applies to 4, and then the linear correction \(2n-5\) is added. At \(n=1\), \(4^1+2-5=1\). At \(n=2\), \(4^2+4-5=15\). At \(n=3\), \(4^3+6-5=65\). At \(n=4\), \(4^4+8-5=259\). Hence the formula matches all four terms. Option B gives \(4^1-3=1\), but at \(n=2\) it gives 13, not 15. Option C gives \(2^1+4=6\), so it fails immediately. Option D gives \(1^4=1\), \(2^4=16\), and \(3^4=81\), which do not match. The rapid growth of the terms suggests a power such as \(4^n\), while the small adjustment is checked by substitution. Exam cue: do not ignore the added \(2n-5\) part.

Correct answer: B. (112)

Explanation: (a_5=109) and (a_1=5) so the sum is (114). In exams recheck the sum before choosing the final option.

Correct answer: D. (a_n=22-4n)

Explanation: The first term is (18) and the difference is (-4), so (a_n=18+(n-1)(-4)=22-4n). Keep the difference negative in a decreasing sequence.

Correct answer: A. 85

Explanation: Given \(a_n=4n^2-3n\). To find the fifth term, substitute \(n=5\): \(a_5=4(5)^2-3(5)=4\times25-15=100-15=85\). Hence, the correct answer is 85. The value 100 comes from \(4\times5^2\) alone; the term \(-3n\) must also be subtracted. Exam tip: substitute the term number first, evaluate the power next, and then perform multiplication and subtraction.

Correct answer: C. 9th term

Explanation: Given a_n=6n+1 and a_n=55, set 6n+1=55. Thus, 6n=54 and n=9. Therefore, 55 is the 9th term of the sequence. The 8th term would be 6(8)+1=49, so it is not correct. Exam tip: To find the position of a given term, equate the general term a_n to that value and solve for n.

Correct answer: D. 6, 12, 20, 30

Explanation: Direct answer: option D, 6, 12, 20, 30. An explicit rule gives a term directly when we know its position. The first four positions are n = 1, 2, 3, and 4. Substitute each value carefully into aₙ = n² + 3n + 2. For n = 1, a₁ = 1² + 3(1) + 2 = 6. For n = 2, a₂ = 2² + 3(2) + 2 = 4 + 6 + 2 = 12. For n = 3, a₃ = 9 + 9 + 2 = 20. For n = 4, a₄ = 16 + 12 + 2 = 30. Thus D matches every term. A gives 4 as its first value, so it misuses the formula. B gives 5 first and 21 third, so its substitutions are wrong. C has the correct first term but incorrect later calculations. D is the only complete match. Memory cue: for a first-term question, begin with n = 1, not n = 0, unless the question explicitly says otherwise.

Correct answer: A. (a_n=\frac{n+1}{2n+1})

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