nth term

The terms are (7\cdot1^2,7\cdot2^2,7\cdot3^2,\ldots), so (a_n=7n^2). Identify the coefficient of the square pattern.

Given a_n=28-5n, a_7=28-5(7)=28-35=-7 and a_{12}=28-5(12)=28-60=-32. Therefore, a_7+a_{12}=-7+(-32)=-39. Option A can result from substituting an incorrect value of n in one of the terms. Exam tip: evaluate each required term separately before adding them.

Write the terms as cubes: \\(64=4^3\\), \\(125=5^3\\), \\(216=6^3\\), and \\(343=7^3\\). The cube bases begin at 4 and increase by 1 for each next term. Therefore, when the term number is \\(n\\), its base is \\(n+3\\): at \\(n=1\\), this gives 4; at \\(n=2\\), it gives 5; and so forth. Thus the nth term is \\(a_n=(n+3)^3\\), so option C is correct.

Substitution verifies the rule: \\(a_1=(1+3)^3=4^3=64\\), \\(a_2=5^3=125\\), \\(a_3=6^3=216\\), and \\(a_4=7^3=343\\). The expression \\(n^3+63\\) has no continuing cube pattern, \\((n+2)^3\\) starts with \\(3^3\\), and \\(4n^3\\) does not produce the listed terms. The consistent starting base and unit increase establish the supplied answer.

The index increases by (6) and the common difference is (10), so the difference is (60). In a linear rule, the coefficient of (n) gives the common difference.

Each term is multiplied by (3), so (a_n=8\cdot3^{n-1}). In a geometric sequence, the first term stays separate.

Given \(a_n=5n^2+2n-9\), \(a_{n+2}=5(n+2)^2+2(n+2)-9=5n^2+22n+15\). Hence, \(a_{n+2}-a_n=(5n^2+22n+15)-(5n^2+2n-9)=20n+24\). The option \(20n+22\) results from an error in calculating the constant term. Exam tip: first expand \((n+2)^2=n^2+4n+4\) carefully.

From (9d=108), (d=12), so (a_{26}=181+9\cdot12=289). It is easier to find a distant term from a nearby given term.

The consecutive differences are \(12-2=10\), \(30-12=18\), \(56-30=26\), and \(90-56=34\). Their second differences are constantly \(8\), so the sequence is quadratic with coefficient \(4\) for \(n^2\). Using \(a_n=4n^2-2n\) gives \(a_1=2\), \(a_2=12\), and \(a_3=30\). Option B gives \(10\) as the second term, so it is not correct. Exam tip: verify a proposed nth-term formula using at least the first three terms.

Given \(a_n=7n-13\). To find \(a_{4n+1}\), substitute the complete index \(4n+1\) for \(n\): \(a_{4n+1}=7(4n+1)-13=28n+7-13=28n-6\). Hence, \(28n-6\) is correct. In \(28n-13\), the effect of the \(+1\) in the index has been omitted. Exam tip: always put a compound subscript in brackets before substituting it into the nth-term formula.

This is an arithmetic progression with first term \(a=23\) and common difference \(d=15\). Its \(n\)th term is \(a_n=a+(n-1)d\). So, \(23+(n-1)\times15=218\) gives \((n-1)\times15=195\), hence \(n-1=13\) and \(n=14\). Therefore, 218 is the 14th term. The 13th term is \(203\), not 218. Exam tip: While finding a term number, remember that the formula contains \(n-1\).

Putting \(n=12\), we get \(a_{12}=\frac{12(12+7)}{4}=\frac{12\times19}{4}=3\times19=57\). Hence, 57 is correct. A value such as 60 can result from incorrectly evaluating \(12+7\) or making an error in multiplication or division. Exam tip: simplify 12 and 4 first, then multiply by 19.

The second differences are (8), and (4n^2+8n-1) gives all starting terms. In a quadratic sequence, test options on the first three terms.

Substituting \(n=6\), \(a_6=9\cdot2^{6-1}+4=9\cdot2^5+4=9\cdot32+4=292\). Hence, the correct answer is 292. The value \(290\) can result from an incorrect addition of the final \(+4\). Exam tip: in exponential expressions, calculate \(n-1\) first and then evaluate the power.

For odd (n), terms are negative and for even (n), terms are positive, so (a_n=4(-1)^n n). In alternating signs, test first at (n=1).

Given \(a_n=n^2+14\), set the term equal to 183: \(n^2+14=183\). Thus, \(n^2=169\), so \(n=13\), since a term number must be a positive integer. Therefore, 183 is the 13th term. For example, the 12th term is \(12^2+14=158\), not 183. Exam tip: To find the position of a given term, equate \(a_n\) to that value and solve for \(n\).

Each term is halved, so \(a_n=48\left(\frac{1}{2}\right)^{n-1}\). In a geometric sequence, the exponent starts with (n-1).

Given \(a_n=6n^2-11n+9\), \(a_{11}=6(11)^2-11(11)+9=614\) and \(a_5=6(5)^2-11(5)+9=104\). Therefore, \(a_{11}-a_5=614-104=510\). The value 540 results from incorrectly calculating \(a_5\) as 74. Exam tip: substitute each value of \(n\) carefully and find both terms separately before subtracting.

The first term is (27) and the difference is (22), so (a_{20}=27+19\cdot22=445). The (20)th term includes (19) differences.

The governing concept is evaluating an explicit sequence rule at two specified indices and then subtracting. First calculate a₈: a₈ = 4(8²) + 7(8) = 4(64) + 56 = 256 + 56 = 312. Next calculate a₃: a₃ = 4(3²) + 7(3) = 4(9) + 21 = 36 + 21 = 57. Therefore, a₈ − a₃ = 312 − 57 = 255, so option B is correct. Option A, C, and D can result from arithmetic slips such as miscalculating one square, multiplying incorrectly, or subtracting inaccurately. The original answer marking was corrected because 255, not 265, follows from the stated rule.

The terms are (9\cdot1^2,9\cdot2^2,9\cdot3^2,\ldots), so (a_n=9n^2). Identify the coefficient of the square pattern.