nth term

Correct answer: D. 20th

Explanation: To find the term number for 131, put a_n=131. Then 7n-9=131, so 7n=140 and n=20. Therefore, 131 is the 20th term. The 19th term is 7(19)-9=124, so it is not correct. Exam tip: To find the position of a given term, substitute its value for a_n and solve for n.

Correct answer: B. 4

Explanation: The term can be written as
a_n=n^2-6n+13=(n-3)^2+4
. Since the minimum value of
(n-3)^2
is 0, attained at n=3, the smallest term value is 4. Option 3 is a common confusion because it is the value of n, not the value of the term. Exam tip: rewrite a quadratic in completed-square form to find its minimum quickly.

Correct answer: C. \(\frac{n}{n+2}\)

Explanation: The numerators \(1,4,9,16\) are \(1^2,2^2,3^2,4^2\), so the numerator of the nth term is \(n^2\). The denominators \(3,8,15,24\) are respectively \(1(1+2),2(2+2),3(3+2),4(4+2)\). Hence, the denominator is \(n(n+2)\). Therefore, the nth term is \(\frac{n^2}{n(n+2)}=\frac{n}{n+2}\). Option B has the numerator pattern but not the required denominator. Exam tip: in fractional sequences, find the numerator and denominator patterns separately before simplifying.

Correct answer: B. 133

Explanation: Substituting \(n=4\) into the formula gives \(a_4=5\cdot3^{4-1}-2=5\cdot3^3-2=5\cdot27-2=133\). Therefore, 133 is correct. A value such as 131 can result from an error in evaluating \(3^3\) or in the subsequent multiplication. Exam tip: in expressions with exponents, simplify the exponent first, then evaluate the power and multiplication.

Correct answer: D. \(2n^2-3n+5\)

Explanation: The first differences are \(3,7,11,15\), and the second differences are all \(4\). Hence, the nth term has the quadratic form \(an^2+bn+c\). Since \(2a=4\), we get \(a=2\). Using the first two terms gives \(b=-3\) and \(c=5\), so the nth term is \(2n^2-3n+5\). Option B has the same leading coefficient, but although it gives 4 for \(n=1\), it gives 8 for \(n=2\), not 7. Exam tip: when second differences are constant, assume a quadratic expression and determine its coefficients from initial terms.

Correct answer: C. 6th term

Explanation: We need \(4n^2-1>100\). Thus, \(4n^2>101\), so \(n^2>25.25\). At \(n=5\), \(a_5=99\), which is not greater than 100, whereas \(a_6=143\). Therefore, the first term greater than 100 is the 6th term. Exam tip: after solving an inequality, check the preceding integer term to confirm that the required term is the first one.

Correct answer: D. \(3n^2-3n+1\)

Explanation: Here, \(a_{n-1}=(n-1)^3\). Therefore, \(a_n-a_{n-1}=n^3-(n-1)^3=n^3-(n^3-3n^2+3n-1)=3n^2-3n+1\). Hence, option D is correct. Option A is the result of \((n+1)^3-n^3\), not of the given difference. Exam tip: for the previous term, always replace \(n\) with \(n-1\).

Correct answer: B. 68

Explanation: Given \(a_n=2n^2+kn+3\), put \(n=3\): \(a_3=2(3)^2+3k+3=21+3k\). Since \(a_3=30\), \(21+3k=30\), so \(k=3\). Now put \(n=5\): \(a_5=2(5)^2+3(5)+3=50+15+3=68\). Hence, 68 is correct. A value such as 72 may result from using an incorrect value of \(k\) or making an addition error. Exam tip: first use the given term to find the unknown constant, then substitute the required value of \(n\).

Correct answer: C. \(4n^2+2n\)

Explanation: The consecutive differences are \(14,22,30,38\), and their common second difference is \(8\). Hence the term has the quadratic form \(an^2+bn+c\), where \(2a=8\), so \(a=4\). Using the first two terms gives \(4n^2+2n\). It gives \(6\) for \(n=1\) and \(20\) for \(n=2\). Option B has the correct leading coefficient but incorrect linear and constant terms. Exam tip: for a constant second difference, the coefficient of \(n^2\) is half the second difference.

Correct answer: A. Arithmetic progression

Explanation: In an arithmetic progression, the difference between consecutive terms remains constant: \(a_{n+1}-a_n=d\). Hence, every new term is formed by adding the same fixed number \(d\) to the preceding term. A geometric progression has a constant ratio rather than a constant difference. Exam tip: subtract consecutive terms; if the difference is constant, it is an AP.

Correct answer: D. (\frac{4n+1}{5n+1})

Explanation: The numerator (5,9,13,\ldots) gives (4n+1), and the denominator (6,11,16,\ldots) gives (5n+1). Make separate rules for both sequences.

Correct answer: B. 96

Explanation: Given \(a_n=n^2+2n+5\), \(a_9=9^2+2(9)+5=81+18+5=104\), while \(a_1=1^2+2(1)+5=8\). Hence, \(a_9-a_1=104-8=96\). The option 98 can result from an error while evaluating \(a_1\). Exam tip: substitute the value of \(n\) separately in each term before taking their difference.

Correct answer: C. \(66\)

Explanation: Interpret the given linear term as \(a_n=an+b\). Then \(a_2+a_9=11a+2b=54\) and \(a_4+a_{11}=15a+2b=78\). Subtracting the first equation from the second gives \(4a=24\), so \(a=6\). Using \(11(6)+2b=54\), we get \(b=-6\). Hence, \(a_{12}=12(6)-6=66\). The nearby option \(63\) does not follow from this linear expression. Exam tip: subtract such pair-sum equations first to eliminate the constant term \(b\).

Correct answer: A. \(2n^2+5n+3\)

Explanation: The first differences are \(11,15,19,23\), and the second differences are all \(4\). Hence the sequence has a quadratic nth term, with coefficient of \(n^2\) equal to \(4/2=2\). Let \(a_n=2n^2+bn+c\). Using \(n=1\) and \(n=2\) gives \(b=5\) and \(c=3\), so \(a_n=2n^2+5n+3\). Option B has the same quadratic coefficient but does not fit the second term. Exam tip: for a constant second difference, use a quadratic form for the nth term.

Correct answer: D. 11th term

Explanation: Given \(a_n=100-7n\), a term smaller than 30 must satisfy \(100-7n<30\). This gives \(-7n<-70\), so \(n>10\). The smallest natural-number value of \(n\) is 11; therefore, the 11th term is the first term less than 30. The 10th term is \(a_{10}=30\), which is not less than 30. Exam tip: for “less than,” use \(<\); an equal value is not included.

Correct answer: B. \(\frac{33}{5}\)

Explanation: For \(a_4\), substitute \(n=4\) into the given formula: \(a_4=\frac{2(4)^2+1}{4+1}=\frac{2\times16+1}{5}=\frac{33}{5}\). Therefore, \(\frac{33}{5}\) is correct. The option \(7\) is incorrect because \(7=\frac{35}{5}\), whereas the correct numerator is \(2\times16+1=33\). Exam tip: after substituting the term number, evaluate the power first, followed by multiplication and addition.

Correct answer: C. \(2^{n+1}-2\)

Explanation: The terms can be written as \(4-2, 8-2, 16-2, 32-2, 64-2\). The subtracted value is always \(2\), and the nth term of \(4,8,16,32,64\) is \(2^{n+1}\). Hence, the nth term is \(2^{n+1}-2\). Option \(2^{n+2}-6\) gives the first term correctly, but for \(n=2\) it gives \(10\), not \(6\). Exam tip: verify a proposed nth-term formula by substituting \(n=1\) and \(n=2\).

Correct answer: A. 3

Explanation: Given \(a_n=n^2+mn+4\), we get \(a_2=2^2+2m+4=8+2m\) and \(a_4=4^2+4m+4=20+4m\). Hence, \(a_2+a_4=28+6m=46\), so \(6m=18\) and \(m=3\). If \(m=4\), the sum would be 52, so it is not correct. Exam tip: substitute the value of \(n\) carefully in every term before combining like terms.

Correct answer: A. \(5n^2-5n+3\)

Explanation: The first differences are \(10,20,30,40\), and their second differences are constantly \(10\). Therefore, the sequence has a quadratic nth term whose \(n^2\) coefficient is \(10/2=5\). Substituting \(n=1,2,3\) in \(a_n=5n^2-5n+3\) gives \(3,13,33\), so it is correct. In option B, the sign of the linear term is incorrect. Exam tip: for a constant second difference, the leading coefficient of a quadratic term is half the second difference.

Correct answer: B. \(5n-3\)

Explanation: For \(a_n=5n-3\), \(a_{n+1}-a_n=[5(n+1)-3]-(5n-3)=5\), which is constant for every \(n\). Therefore, it represents an arithmetic progression. The consecutive differences of \(n^2+1\) and \(n(n+1)\) vary, while \(2^n\) is a geometric-type sequence. Exam tip: if \(a_n\) can be written as \(pn+q\), its common difference is \(p\).

Correct answer: C. 31

Explanation: Given \(a_n=4n+(-1)^{n+1}\). Substituting \(n=8\), \(a_8=4\times 8+(-1)^{8+1}=32+(-1)^9=32-1=31\). Hence, 31 is correct. The value 33 would result from incorrectly taking \((-1)^9\) as \(+1\); an odd power of \(-1\) is \(-1\). Exam tip: an even power of \(-1\) is \(+1\), while an odd power is \(-1\).

Correct answer: A. \(\frac{3n-1}{7n+2}\)

Explanation: The numerators \(2,5,8,11,\ldots\) form an arithmetic progression whose nth term is \(2+(n-1)\times3=3n-1\). The denominators \(9,16,23,30,\ldots\) also form an arithmetic progression whose nth term is \(9+(n-1)\times7=7n+2\). Hence, the nth term is \(\frac{3n-1}{7n+2}\). Option D has the correct numerator, but for \(n=1\) its denominator is 10 instead of 9. Exam tip: in a sequence of fractions, identify the numerator and denominator patterns separately.

Correct answer: D. 110

Explanation: Given \(a_n=2n^3-n\), \(a_4=2(4)^3-4=128-4=124\) and \(a_2=2(2)^3-2=16-2=14\). Therefore, \(a_4-a_2=124-14=110\). A nearby choice such as 108 can result from an error in evaluating the cube or subtracting. Exam tip: find each required term separately before taking their difference.

Correct answer: B. \(6n^2-5n+4\)

Explanation: The first differences are \(13,25,37,49\). Their second differences are all \(12\), so the nth term is quadratic and the coefficient of \(n^2\) is \(12/2=6\). Substituting \(n=1,2,3\) in option B gives \(5,18,43\), respectively; hence \(T_n=6n^2-5n+4\). Although option C gives \(5\) when \(n=1\), it gives \(19\) when \(n=2\), so it is incorrect. Exam tip: when second differences are constant, assume the term has the form \(an^2+bn+c\).

Correct answer: C. (6n-2)

Explanation: To calculate the difference, replace n by n+1 in the rule. We get \(a_{n+1}=3(n+1)^2-5(n+1)+7\). Expanding gives \(3n^2+6n+3-5n-5+7=3n^2+n+5\). Now subtract \(a_n=3n^2-5n+7\): \(a_{n+1}-a_n=(3n^2+n+5)-(3n^2-5n+7)=6n-2\).

Therefore option C, \(6n-2\), is correct. The square term contributes a changing amount, so the difference is not a constant. Care is needed when subtracting the whole expression: subtracting \(-5n\) contributes \(+5n\), and subtracting 7 contributes \(-7\). Options A and B lose part of this calculation, while option D does not represent the full difference. The supplied answer is correct.