Explicit or general rule

Correct answer: B. 66

Explanation: The general term is \(a_n=2^n+2\). Substituting \(n=6\), we get \(a_6=2^6+2=64+2=66\). Although 64 is a close distractor, it is only the value of \(2^6\); the given \(+2\) must also be added. Exam tip: substitute the term number for \(n\), evaluate the power first, and then perform the remaining operations.

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

Explanation: Substituting \(n=1,2,3,4\) in \(a_n=3^n+1\) gives \(4,10,28,82\), respectively. Hence, the correct general term is \(a_n=3^n+1\). Although \(a_n=n^3+3\) gives the first term as 4, its second term is 11, not 10. Exam tip: verify a proposed general term using at least the first two or three terms.

Correct answer: C. 82

Explanation: Given \(a_n=3^n+1\), substitute \(n=4\): \(a_4=3^4+1=81+1=82\). Option 81 is only the value of \(3^4\); it misses the added 1 in the rule. Exam tip: after substituting the index in a general term, include both the power and the constant term.

Correct answer: B. (a_n=6\cdot2^n)

Explanation: (6\cdot2^n) gives (12,24,48,96). In exams check both the base and multiplier in geometric growth.

Correct answer: C. 192

Explanation: Given \(a_n=3\cdot2^{n-1}\). Substituting \(n=7\), \(a_7=3\cdot2^{7-1}=3\cdot2^6=3\cdot64=192\). The value 96 can result from incorrectly using \(2^5\). Exam tip: substitute the term number into the exponent \(n-1\) before calculating.

Correct answer: D. (a_n=3\cdot2^{n-1})

Explanation: The first term is (3) and each term is multiplied by (2) so (a_n=3\cdot2^{n-1}). In exams check both the first term and ratio.

Correct answer: A. \(3n+1\)

Explanation: This is an arithmetic sequence because each term increases by 3. Checking \(n=1\), \(3(1)+1=4\), and the common difference remains 3. In contrast, \(3n+4\) gives 7 as its first term. Exam tip: verify both the first term and common difference.

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

Explanation: The governing concept is selecting an explicit rule by testing the pattern of differences. The first differences are 13−7=6, 23−13=10, and 37−23=14; these increase by 4, so the second difference is constant. A quadratic rule is therefore plausible. Test option A: for n=1, 2(1)^2+5=7; for n=2, 2(2)^2+5=13; for n=3, 2(3)^2+5=23; and for n=4, 2(4)^2+5=37. Thus option A reproduces every given term and is correct. Option B is linear and gives 14 at n=1, option C gives 7 initially but 10 at n=2, and option D gives 7 initially but has constant difference 6, so none matches the sequence.

Correct answer: B. 30:7

Explanation: Given a_n=4n²−n, a_4=4(4²)−4=64−4=60 and a_2=4(2²)−2=16−2=14. Hence, a_4:a_2=60:14=30:7. Option A uses the correct terms but does not simplify the ratio. Exam tip: after finding a ratio, divide both terms by their greatest common factor to write it in simplest form.

Correct answer: A. \(a_2=15,\ a_3=32\)

Explanation: Given \(a_n=3n^2+2n-1\). Substituting \(n=2\), \(a_2=3(2)^2+2(2)-1=12+4-1=15\). Similarly, for \(n=3\), \(a_3=3(3)^2+2(3)-1=27+6-1=32\). Hence, option A is correct. Option B has an incorrect value of \(a_2\), while option C has an incorrect value of \(a_3\). Exam tip: substitute the value of \(n\) separately for each term and calculate the square carefully.

Correct answer: A. \(a_n=3n^2+2n-1\)

Explanation: The consecutive differences are \(11,17,23\), and their second differences are constant at \(6\). Hence, the sequence has a quadratic general term. Substituting \(n=1,2,3,4\) in \(a_n=3n^2+2n-1\) gives \(4,15,32,55\), respectively. The option \(a_n=11n-7\) matches only the first two terms, not the later terms. Exam tip: verify a proposed general term using at least the first three terms.

Correct answer: A. (42)

Explanation: The first three terms are (5,14,27) and the sum is (46). In exams write all terms before adding.

Correct answer: D. 42

Explanation: Given \(a_n=6n-4\), \(a_{12}=6(12)-4=68\) and \(a_5=6(5)-4=26\). Therefore, \(a_{12}-a_5=68-26=42\). Option 40 may result from an error in finding a term or subtracting the values. Exam tip: substitute the value of \(n\) to find each term separately, then subtract.

Correct answer: C. (12)th term

Explanation: Its rule is (a_n=6n-4) and (6n-4=68) gives (n=12). In exams equate the given term to the general term.

Correct answer: B. 3

Explanation: Given \(a_n=\frac{2n+3}{5}\). Substituting \(n=6\), \(a_6=\frac{2(6)+3}{5}=\frac{12+3}{5}=\frac{15}{5}=3\). Hence, 3 is the correct answer. \(\frac{13}{5}\) may result from an error in calculating \(2\times6\). Exam tip: substitute the value of \(n\) first, then simplify step by step.

Correct answer: B. aₙ = (2n + 3)/5

Explanation: The governing concept is an explicit rule that generates each term directly from its position n. Test option B at the first four positions. When n = 1, (2n + 3)/5 = 5/5 = 1. When n = 2, it equals 7/5; when n = 3, it equals 9/5; and when n = 4, it equals 11/5. Thus option B reproduces every displayed term exactly. Option A gives 1 for the first term but gives 6/5 for the second, not 7/5. Option C gives 7/3 at n = 1, and option D produces 1, 3, 5, 7. Checking several positions is necessary because matching only the first term cannot establish a valid general rule.

Correct answer: B. 82

Explanation: Given \(a_n=10n+1\), \(a_2=10\times2+1=21\) and \(a_6=10\times6+1=61\). Therefore, \(a_2+a_6=21+61=82\). Option 84 would result from calculating one of the terms incorrectly. Exam tip: substitute the value of \(n\) in the general term and find each required term separately.

Correct answer: B. (a_n=10n+1)

Explanation: The direct answer is option B: \(a_n=10n+1\). Start with the meaning of a general term: \(n\) tells the position, so the rule must give 11 when \(n=1\), 21 when \(n=2\), and so on. The sequence increases by 10 each time, so the coefficient of \(n\) is 10. For the first term, \(10(1)=10\), and 1 must be added to obtain 11; hence \(a_n=10n+1\). Checking: at \(n=2\), 21; at \(n=3\), 31. Option A, \(11n\), gives 11 first but 22 second, so its difference is wrong. Option B gives every listed term and is correct. Option C, \(n+10\), gives 11 first but then 12, so it increases by only 1. Option D, \(10n-1\), gives 9 first, not 11. Memory cue: for a linear sequence, use difference times position, then adjust the constant using the first term.

Correct answer: C. 102

Explanation: Given \(a_n=4n^2+1\), \(a_3=4(3)^2+1=37\) and \(a_4=4(4)^2+1=65\). Therefore, \(a_3+a_4=37+65=102\). An answer such as 100 can result from an error in squaring or addition. Exam tip: Substitute each value of \(n\) separately in the general term before adding.

Correct answer: A. \(a_n=4n-1\)

Explanation: An explicit rule gives any term directly from its position \(n\). \(a_n=4n-1\) contains only \(n\), whereas the second rule needs the previous term. Exam tip: the presence of \(a_{n-1}\) usually indicates a recursive rule.

Correct answer: A. aₙ = 2ⁿ − n

Explanation: The governing idea is to test an explicit formula at the corresponding positive integer positions. For option A, substitute n=1, 2, 3, and 4: 2¹−1=1, 2²−2=2, 2³−3=5, and 2⁴−4=12. These values reproduce exactly the four displayed terms, so option A is the correct general rule. Option B produces the odd-number pattern 1, 3, 5, 7. Option C produces square numbers 1, 4, 9, 16. Option D produces 1, 3, 7, 15. Some alternatives agree with the first term, but a valid general rule must generate all the given terms at their correct positions. Only A satisfies that requirement, including the less obvious fourth term.

Correct answer: B. 21

Explanation: Given \(a_n=3^n-2n\). Substituting \(n=3\), \(a_3=3^3-2(3)=27-6=21\). Therefore, 21 is correct. A result such as 19 would come from an error in evaluating the power or subtraction. Exam tip: substitute the value of \(n\) everywhere before simplifying the expression.

Correct answer: C. aₙ = 3ⁿ − 2n

Explanation: Use the explicit-rule principle: substitute each position n into a candidate and compare the results with the sequence. For option C, n=1 gives 3¹−2(1)=1; n=2 gives 3²−2(2)=9−4=5; n=3 gives 3³−6=27−6=21; and n=4 gives 3⁴−8=81−8=73. Thus aₙ=3ⁿ−2n reproduces every listed term, making option C correct. Option A gives 2 at n=1, so it fails immediately. Option B is linear and gives 4 at n=2. Option D gives 3 at n=1 and 6 at n=2. The term −2n is essential: using only 3ⁿ or using a linear expression cannot account for the stated values.

Correct answer: B. \(a_5\)

Explanation: Given \(a_n=n^3-1\), set \(a_n=124\). Then \(n^3-1=124\), so \(n^3=125=5^3\). Hence \(n=5\), and therefore \(a_5=124\). The close distractor \(a_4\) is incorrect because \(a_4=63\). Exam tip: for expressions of the form \(n^3-1\), add 1 first and identify the perfect cube.

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

Explanation: For \(a_n=n^3-1\), we get \(a_1=1^3-1=0\), \(a_2=2^3-1=7\), \(a_3=3^3-1=26\), and \(a_4=4^3-1=63\). Hence, the correct general term is \(a_n=n^3-1\). The nearby distractor \(a_n=n^2-1\) gives \(8\) as its third term, not \(26\). Exam tip: substitute at least the first three values of \(n\) to verify a proposed general term.