Explicit or general rule

Substitute n=3: a_3=4^3-3^2=64-9=55. Therefore, 55 is the correct option. An answer such as 53 may result from an error in evaluating the power or the square. Exam tip: substitute the value of n in every part of the rule before calculating powers and squares.

(4^n-n^2) gives (3,12,55,240). In exams, also check rules where a square is subtracted from a power.

The first five terms are (5,17,29,41,53), and the average is (29). In exams, divide the sum by the number of terms.

Its rule is (a_n=12n-7), and (12n-7=149) gives (n=13). In exams, equate the given term to the general term.

Substituting \(n=2\) into the rule gives \(a_2=6(2)^2-4(2)+3=24-8+3=19\). Similarly, for \(n=3\), \(a_3=6(3)^2-4(3)+3=54-12+3=45\). 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 each value of \(n\) separately and simplify carefully.

The governing concept is identifying an explicit rule from finite differences. The first differences are 19−5 = 14, 45−19 = 26, and 83−45 = 38. The second differences are 12 and 12, so a quadratic rule is appropriate. Substituting n = 1, 2, 3, 4 into option A gives 6−4+3 = 5, 24−8+3 = 19, 54−12+3 = 45, and 96−16+3 = 83. Therefore A matches every given term. Option B and D are linear, while option C gives 5 at n = 1 but 26 at n = 2, not 19.

Given \(a_n=5^n-2n\). Substituting \(n=3\), \(a_3=5^3-2(3)=125-6=119\). Therefore, 119 is correct. The value 121 may result from incorrectly subtracting 4 instead of \(2n=6\). Exam tip: substitute the required term number for \(n\) in every part of the rule.

Given \(a_n=2n^2+7n-4\), substitute \(n=6\): \(a_6=2(6)^2+7(6)-4=2\times36+42-4=72+42-4=110\). Therefore, the correct answer is 110. The value 108 may result from an error while calculating \(7\times6\). Exam tip: substitute the term number carefully and evaluate the power term first.

The correct general term is \(a_n=2n^2+7n-4\). Substituting \(n=1\), \(2\), and \(3\) gives \(a_1=5\), \(a_2=18\), and \(a_3=35\), respectively. The distractor \(13n-8\) may match the first two terms, but for \(n=3\) it gives \(31\), not \(35\). Exam tip: verify a proposed general term using at least the first three terms.

Substitute \(n=6\): \(a_6=\frac{2(6)^2+5(6)}{3}=\frac{2\times36+30}{3}=\frac{72+30}{3}=\frac{102}{3}=34\). Hence, 34 is the correct option. An answer such as 36 can result from not dividing the complete numerator correctly by 3. Exam tip: substitute the value of \(n\) first, then follow the order of operations—powers, multiplication, addition, and finally division.

The governing concept is evaluating an explicit formula at two indices and then subtracting in the stated order. First, a_4 = 4(4^2) + 3(4) + 2 = 4(16) + 12 + 2 = 78. Next, a_1 = 4(1^2) + 3(1) + 2 = 4 + 3 + 2 = 9. Therefore a_4 − a_1 = 78 − 9 = 69, so option B is correct. Option A can result from an arithmetic error in the quadratic term, while C and D arise from incorrect substitution or subtraction. Computing both sequence values separately helps preserve the required order.

(4n^2+3n+2) gives (9,24,47,78). In exams, check a quadratic rule when second differences are constant.

Substitute \(n=5\): \(a_5=3\cdot2^5-5=3\cdot32-5=96-5=91\). Therefore, the correct answer is 91. A value such as 89 can result from subtracting the wrong value instead of 5. Exam tip: evaluate the exponent first, then multiply, and finally subtract \(n\).

Substituting \(n=1,2,3,4\) into \(a_n=3\cdot2^n-n\) gives \(5,10,21,44\), respectively. Hence, option A is correct. Option C gives 3 when \(n=1\), so it does not even produce the first term. Exam tip: verify an explicit rule by checking at least the first three terms.

(a_5=99) and (a_2=8), so the sum is (107). In exams, handle signs carefully in a quadratic formula.

The first differences are \(14,22,30\), and the second differences are constant at \(8\). Hence the sequence is quadratic, with coefficient of \(n^2\) equal to \(8/2=4\). Substituting \(n=1,2,3,4\) in \(a_n=4n^2+2n\) gives \(6,20,42,72\), respectively. Option B matches the first term but gives \(16\), not \(20\), when \(n=2\). Exam tip: verify a proposed general term using at least the first three terms.

The first term is (5) and the common difference is (7), so (a_n=5+(n-1)7=7n-2). In an arithmetic sequence the coefficient is the common difference.

The first term is (42) and the difference is (-6), so (a_n=42+(n-1)(-6)=48-6n). Keep the difference negative for a decreasing sequence.

The rule is \(a_n=4n^2-5n+2\). Substituting \(n=6\), \(a_6=4(6)^2-5(6)+2=4\times36-30+2=116\). Therefore, 116 is the correct answer. A value such as 112 can result from an error while evaluating the squared term. Exam tip: calculate \(n^2\) first, then perform multiplication and the remaining operations.

Given \(a_n=9n+4\) and \(a_n=112\), set \(9n+4=112\). This gives \(9n=108\), so \(n=12\). Therefore, 112 is the 12th term of the sequence. The 11th term is \(9(11)+4=103\), 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 \(n\).