Explicit or general rule

Correct answer: A. \(a_n=5n-2\)

Explanation: For \(a_n=5n-2\), \(a_{n+1}-a_n=[5(n+1)-2]-(5n-2)=5\), which is constant, so it is an AP. In \(n^2+1\), the difference changes. Exam tip: check consecutive-term differences.

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

Explanation: The correct rule is \(a_n=2n^3+n\). Checking it: for \(n=1\), \(a_1=2(1)^3+1=3\); for \(n=2\), \(a_2=2(2)^3+2=18\); and for \(n=3\), \(a_3=2(3)^3+3=57\). Thus, it matches the given sequence. In contrast, \(3n^2\) gives 12 as the second term, not 18. Exam tip: substitute at least the first two or three values of \(n\) to verify a proposed general term.

Correct answer: B. (94)

Explanation: (a_4=56) and (a_7=38) so the sum is (94). In exams find both terms carefully in a decreasing formula.

Correct answer: B. (a_n=80-6n)

Explanation: At (n=1) it gives (74) and at (n=2) it gives (68) so (a_n=80-6n). In exams check the first two terms of a decreasing sequence.

Correct answer: C. 12

Explanation: To find \(a_4\), substitute \(n=4\) in the general-term formula: \(a_4=\frac{4(2\times4+1)}{3}=\frac{4\times9}{3}=12\). Therefore, the correct answer is 12. Values such as 10 or 11 may result from evaluating \(2n+1\) incorrectly. Exam tip: substitute the required term number carefully before simplifying.

Correct answer: A. aₙ = \frac{n(2n+1)}{3}

Explanation: To verify an explicit rule, substitute consecutive values of n and simplify each expression carefully. For option A, n=1 gives 1(2+1)/3=1; n=2 gives 2(4+1)/3=10/3; n=3 gives 3(6+1)/3=7; and n=4 gives 4(8+1)/3=12. This agrees with every displayed term, so option A is correct. Option B generates 1, 3, 6, 10, the triangular-number sequence. Option C generates 1, 3, 5, 7, while option D generates 2, 7, 15, 26. The fractional second term is a useful discriminator: a rule that ignores the denominator or changes the growth pattern cannot be accepted, even if its first term happens to be 1.

Correct answer: C. 61

Explanation: Given \(a_n=4^n-n\). Substituting \(n=3\), we get \(a_3=4^3-3=64-3=61\). Therefore, 61 is the correct option. The value 60 would result from subtracting 4 from 64, but the formula requires subtracting \(n=3\). Exam tip: substitute the term number first, then evaluate the exponent carefully.

Correct answer: D. \(a_n=2n^2+3n\)

Explanation: The correct rule is \(a_n=2n^2+3n\). Substituting \(n=1,2,3,4\) gives \(5,14,27,44\), respectively. Option A gives the first term as 5, but its second term is \(12\), so it does not fit the sequence. Exam tip: Always test a proposed general term for at least the first three terms.

Correct answer: B. aₙ = (n + 1)³

Explanation: The governing concept is an explicit or general rule: a formula must produce the term directly from its position n. Rewrite the sequence as 2³, 3³, 4³, 5³, ... . The first displayed term corresponds to n = 1, so its cube base is n + 1; therefore the rule is aₙ = (n + 1)³. Checking confirms it: for n = 1, a₁ = 2³ = 8; for n = 2, a₂ = 3³ = 27; for n = 3, a₃ = 4³ = 64; and for n = 4, a₄ = 5³ = 125. Thus option B is correct. Option A starts with 1³, option C gives 2 at n = 1, and option D starts with 3³, so those alternatives do not match the sequence.

Correct answer: B. 45

Explanation: The governing concept is substitution into an explicit sequence rule, followed by subtraction of two terms. For the eighth term, substitute n = 8: a₈ = 9(8) − 4 = 72 − 4 = 68. For the third term, substitute n = 3: a₃ = 9(3) − 4 = 27 − 4 = 23. Therefore a₈ − a₃ = 68 − 23 = 45, so option B is correct. A useful shortcut is that the constant term cancels: (9 × 8 − 4) − (9 × 3 − 4) = 9(8 − 3) = 45. Option A could result from an arithmetic error, while options C and D do not follow from the required evaluations. Both terms must be calculated using their correct indices before finding the difference.

Correct answer: B. aₙ = 5n + 3

Explanation: The governing concept is finding an explicit rule for an arithmetic sequence. Consecutive terms increase by a constant difference: 13 − 8 = 5, 18 − 13 = 5, and 23 − 18 = 5. An arithmetic sequence has aₙ = a₁ + (n − 1)d, so here aₙ = 8 + (n − 1)5 = 8 + 5n − 5 = 5n + 3. Thus option B is correct. Verification is important: at n = 1, the formula gives 5(1) + 3 = 8; at n = 2 it gives 13; at n = 3 it gives 18; and at n = 4 it gives 23. Option A gives 8, 16, 24, ... and has the wrong difference. Option C begins with 2, and option D has difference 3, so neither describes the given sequence.

Correct answer: A. \(a_n=5n-2\)

Explanation: \(a_n=5n-2\) has the form \(pn+q\), so \(a_{n+1}-a_n=5\) is constant. For \(n^2+1\), successive differences change. Exam tip: check whether the common difference is constant.

Correct answer: A. (a_n=9n+5)

Explanation: The direct answer is option A: \(a_n=9n+5\). Compare consecutive terms: \(23-14=9\), \(32-23=9\), and \(41-32=9\). Thus this is an arithmetic sequence with common difference 9. A linear rule has the form \(a_n=9n+c\). Using the first term, \(9(1)+c=14\), so \(c=5\); therefore \(a_n=9n+5\). Checking gives 14, 23, 32, and 41 in order. Option A is correct. Option B, \(14n\), gives 14 first but 28 second, so it does not preserve the difference. Option C, \(9n-5\), gives 4 first, not 14. Option D, \(5n+9\), gives the first term 14 but has difference 5, so its second term is 19. Exam cue: identify the common difference first, then use the first term to find the constant.

Correct answer: C. (a_n=76-6n)

Explanation: At (n=1) it gives (70) and at (n=2) it gives (64) so (a_n=76-6n). In exams match the first term in a decreasing sequence.

Correct answer: C. n = 16

Explanation: The governing concept is finding the position of a term from an explicit formula. Set the given rule equal to the target value: 4n + 9 = 73. Subtract 9 from both sides to obtain 4n = 64, then divide by 4 to get n = 16. Therefore 73 is the sixteenth term, and option C is correct. Direct checking gives a₁₆ = 4(16) + 9 = 64 + 9 = 73. The other options produce different values: a₁₄ = 65, a₁₅ = 69, and a₁₇ = 77. A common error is to subtract 9 correctly but divide incorrectly, or to confuse the value of a term with its position. Here n is the position, while 73 is the term value.

Correct answer: C. (9)th term

Explanation: Its rule is (a_n=7n-2) and (7n-2=61) gives (n=9). In exams form the general term first to find the term number.

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

Explanation: The direct answer is option B, \(a_n=(n+3)^2\). The terms are consecutive squares: \(16=4^2\), \(25=5^2\), \(36=6^2\), and \(49=7^2\). For positions \(n=1,2,3,4\), the square roots are respectively \(n+3\), so the rule is \((n+3)^2\). Substitution confirms it: \((1+3)^2=16\), \((2+3)^2=25\), \((3+3)^2=36\), and \((4+3)^2=49\). Option A, \(n^2\), gives 1, 4, 9, 16, so it begins too early. Option C, \(n^2+3\), gives 4, 7, 12, 19; adding 3 after squaring does not create the required shifted squares. Option D, \(4n^2\), gives 4, 16, 36, 64, so it has a different pattern. The key is to find the square root first, then express that root using \(n\).

Correct answer: B. 64

Explanation: The general term is \(a_n=(n+3)^2\). Substituting \(n=5\), we get \(a_5=(5+3)^2=8^2=64\). Hence, 64 is the correct answer. The value 81 would result only if the quantity inside the bracket were 9. Exam tip: evaluate the expression inside brackets before squaring it.

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

Explanation: For \(a_n=n^2+3n+2\), substituting \(n=1,2,3,4\) gives 6, 12, 20, and 30 respectively. Hence, option A is correct. Option B gives 12 as its second term, but its third term is 18, not 20. Exam tip: verify a proposed general term by checking at least the first three terms.

Correct answer: A. \(a_n=3n-1\)

Explanation: In option A, \(a_{n+1}-a_n=[3(n+1)-1]-(3n-1)=3\), which is constant, so it is an arithmetic progression. For \(n^2+1\), the differences change. Exam tip: check consecutive-term differences.

Correct answer: A. (a_n=\frac{n(n+1)}{2}+3)

Explanation: Adding (3) to triangular numbers gives (4,6,9,13). In exams think of triangular numbers when differences are (2,3,4).

Correct answer: C. 31

Explanation: Substitute \(n=7\): \(a_7=\frac{7(7+1)}{2}+3=\frac{7\times8}{2}+3=28+3=31\). Hence, 31 is correct. The close distractor 28 is only the value of \(\frac{7\times8}{2}\); the given \(+3\) must still be added. Exam tip: Substitute the required index for \(n\) and simplify step by step.

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

Explanation: In option A, substituting \(n=1,2,3,4\) gives \(1,6,13,22\), respectively. Therefore, the general term is \(a_n=n^2+2n-2\). Option C matches only the first two terms; at \(n=3\), it gives 11 instead of 13. Exam tip: verify a proposed general term using at least three or four terms.

Correct answer: C. (n=9)

Explanation: Putting (n=9) gives (97), while (n=8) gives (78), so none of the options is correct. In exams check options by direct substitution.

Correct answer: B. 67

Explanation: Given \(a_n=2^n+3\), substitute \(n=6\): \(a_6=2^6+3=64+3=67\). Option 65 could result from an addition error; the constant 3 must be added to 64. Exam tip: to find a term from an explicit rule, substitute the term number first and evaluate the exponent carefully.