nth term

Correct answer: A. (\frac{11n}{6})

Explanation: Each term increases by (\frac{11}{6}), so (a_n=\frac{11n}{6}). Use a common denominator to see the pattern in fractions.

Correct answer: B. (5n^2-1)

Explanation: The terms match (5n^2+1), so none of the listed options is correct. Always test options on the first term.

Correct answer: A. \(16n+1\)

Explanation: Given \(a_n=8n^2-7n+4\), substitute \(n+1\) for \(n\): \(a_{n+1}=8(n+1)^2-7(n+1)+4=8n^2+9n+5\). Therefore, \(a_{n+1}-a_n=(8n^2+9n+5)-(8n^2-7n+4)=16n+1\). The option \(16n-7\) results from incorrectly handling the linear term produced when expanding \((n+1)^2\). Exam tip: find \(a_{n+1}\) completely before subtracting \(a_n\).

Correct answer: B. 235

Explanation: Given \(a_n=2n^3-3n\), substitute \(n=5\): \(a_5=2(5)^3-3(5)=2\times125-15=250-15=235\). Hence, 235 is correct. 250 is only the value of \(2(5)^3\); the term \(3\times5\) must still be subtracted. Exam tip: evaluate the power first, then multiply and subtract.

Correct answer: A. (51-7n)

Explanation: The first term is (44) and the difference is (-7), so (a_n=44+(n-1)(-7)=51-7n). Use a negative difference in a decreasing sequence.

Correct answer: A. \(11-4(n-1)\)

Explanation: For an AP, \(a_n=a_1+(n-1)d\). Substituting \(a_1=11\) and \(d=-4\) gives \(a_n=11-4(n-1)\). In option D, the first term is 7, not 11. Exam tip: always verify the first term first.

Correct answer: A. (2n^2+10n+6)

Explanation: The second differences are (4), and (2n^2+10n+6) gives the starting terms. Confirm a quadratic rule using the first three terms.

Correct answer: C. 15th term

Explanation: Given \(a_n=13n-19\), set the nth term equal to 176: \(13n-19=176\). Thus, \(13n=195\), so \(n=15\). Therefore, 176 is the 15th term of the sequence. The 14th term is \(13(14)-19=163\), so it is not correct. Exam tip: To find a term number, equate \(a_n\) to the given term and solve for n.

Correct answer: C. (\frac{13n}{7})

Explanation: Each term increases by (\frac{13}{7}), so (a_n=\frac{13n}{7}). In fractions with a common denominator, observe the numerator pattern.

Correct answer: B. 150

Explanation: Substituting the values in the given formula gives \(a_9=3(9)^2+5(9)-8=280\) and \(a_6=3(6)^2+5(6)-8=130\). Therefore, \(a_9-a_6=280-130=150\). Option 120 is incorrect because it uses 160 as the value of \(a_6\). Exam tip: Evaluate both terms separately before finding their difference.

Correct answer: A. (7n^2)

Explanation: 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.

Correct answer: B. -39

Explanation: 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.

Correct answer: C. ((n+3)^3)

Explanation: 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.

Correct answer: C. (60)

Explanation: 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.

Correct answer: A. (8\cdot3^{n-1})

Explanation: The correct answer is option A: \\(a_n=8\cdot3^{n-1}\\). Each term is three times the preceding term: \\(24/8=3\\), \\(72/24=3\\), and \\(216/72=3\\). Thus the sequence is geometric, with first term \\(a=8\\) and common ratio \\(r=3\\). Using \\(a_n=ar^{n-1}\\), we get \\(a_n=8\cdot3^{n-1}\\). Option A gives 8 at n=1 and reproduces all listed terms. Option B has first term 3, not 8. Option C uses \\(3^n\\), giving 24 for n=1, so the exponent is too large by one. Option D also begins with 24, so it does not describe the given sequence from its first term. Exam cue: test n=1 first; the correct nth-term formula must produce the first listed term.

Correct answer: A. \(20n+24\)

Explanation: 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.

Correct answer: B. (289)

Explanation: 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.

Correct answer: A. \(4n^2-2n\)

Explanation: 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.

Correct answer: A. \(28n-6\)

Explanation: 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.

Correct answer: C. 14th term

Explanation: 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 \(n-1\).

Correct answer: B. 57

Explanation: 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.

Correct answer: A. (4n^2+8n-1)

Explanation: 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.

Correct answer: C. 292

Explanation: 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.

Correct answer: B. (4(-1)^n n)

Explanation: 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).

Correct answer: C. 13th term

Explanation: 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 \(n\).