nth term

Given \(a_n=18-6n\), substitute \(n=5\) to find the fifth term: \(a_5=18-6(5)=18-30=-12\). Therefore, option C is correct. A nearby distractor such as \(-9\) can result from multiplying \(6\times5\) incorrectly. Exam tip: after substituting the value of \(n\), perform multiplication before subtraction.

The first differences are \(9,11,13\), whose differences are constant at \(2\); hence the pattern is quadratic. For \(a_n=n^2+6n\), substituting \(n=1,2,3,4\) gives \(7,16,27,40\), respectively. \(9n-2\) fits only the first two terms and gives \(25\) for the third term. Exam tip: when second differences are constant, test a rule containing \(n^2\).

Putting \(n=4\), \(a_4=4\cdot5^{4-1}=4\cdot5^3=4\cdot125=500\). Hence, the correct answer is 500. The value 400 would result from incorrectly using an exponent of 2. Exam tip: when substituting into \(a_n\), simplify the exponent \(n-1\) carefully first.

This is an arithmetic progression with first term 17 and common difference 5. Therefore, its nth term is \(a_n=17+(n-1)\times5=5n+12\). Putting \(a_n=87\) gives \(5n+12=87\), so \(5n=75\) and \(n=15\). At position 14, the term would be 82, not 87. Exam tip: first identify the first term and common difference, then use \(a_n=a+(n-1)d\).

Given \(a_n=n^2+4n-2\), substitute \(n=6\): \(a_6=6^2+4\times6-2=36+24-2=58\). Therefore, 58 is correct. The value 60 would result if the final \(-2\) were omitted. Exam tip: after substituting a value in an nth-term rule, follow the order of operations carefully.

The first term of this GP is 3 and the common ratio is 5. Its terms are 3, 15, 75, 375, 1875. Therefore, 1875 is the fifth term. The sixth term would be 1875 × 5 = 9375, so the sixth-term option is not correct. Exam tip: To find a term’s position, list the terms in order or use \(a_n=ar^{n-1}\).

The governing concept is the nth-term formula for an arithmetic progression: a_n = a + (n − 1)d. The first term is a = 9, and the common difference is d = 19 − 9 = 10. Substitution gives a_n = 9 + (n − 1)10 = 9 + 10n − 10 = 10n − 1. Therefore, option A is correct. A quick check confirms it: for n = 1, the formula gives 10(1) − 1 = 9; for n = 2, it gives 19. Option B gives 11 for the first term, while options C and D do not preserve the given constant difference of 10.

To find the fifth term, substitute \(n=5\): \(a_5=\frac{7(5)-2}{3}=\frac{35-2}{3}=\frac{33}{3}=11\). Therefore, 11 is the correct option. Option 10 may result from an error in subtraction or division. Exam tip: In an \(a_n\) formula, substitute the required value of \(n\) and simplify step by step.

The sequence is 4, 16, 36, 64, and each term can be viewed as four times a square. The first term is \\(4\\times1^2\\), the second is \\(4\\times2^2\\), the third is \\(4\\times3^2\\), and the fourth is \\(4\\times4^2\\). Thus the number inside the square is exactly the position number. Recognizing this structure is more reliable than guessing from differences alone.

For the nth position, replace the position number by \\(n\\). The general term is therefore \\(a_n=4n^2\\). Checking it confirms the pattern: \\(4(1)^2=4\\), \\(4(2)^2=16\\), \\(4(3)^2=36\\), and \\(4(4)^2=64\\). Hence option A is correct. Option B gives half the required values, while the linear expression in option C cannot produce this square-based growth.

Given \(a_n=12n+5\), \(a_7=12\times7+5=89\) and \(a_2=12\times2+5=29\). Therefore, \(a_7+a_2=89+29=118\), so option A is correct. A value such as 116 can result from a small arithmetic error while adding or multiplying. Exam tip: evaluate each required term separately before adding them.

The terms are (9,27,81,243,729,2187), so (2187) is the sixth term. In a geometric progression, multiply by the same ratio each time.

To find the sixth term, substitute n=6 in the rule: \(a_6=3(6)^2+5(6)=3\times36+30=108+30=138\). Therefore, 138 is correct. A value such as 128 may result from an error in squaring or multiplication. Exam tip: evaluate the exponent first, then multiply, and finally add.

The first term is (50) and the difference is (-7), so (a_n=50+(n-1)(-7)=57-7n). In a decreasing sequence, take the difference as negative.

To find the fourth term, substitute \(n=4\) in the formula: \(a_4=4+8=12\). Therefore, the correct answer is 12. Option 13 would result from using \(n=5\), so it is the fifth term. Exam tip: the subscript \(n\) in \(a_n\) indicates the term number.

The first term is 7 and the common difference is 4. Thus \(a_n=7+(n-1)\times4=4n+3\). Option B gives 1 when \(n=1\), not 7. Exam tip: verify both the first term and common difference.

For \(n^2+1\), the difference of consecutive terms is \((n+1)^2+1-(n^2+1)=2n+1\), which changes with n. An AP needs a constant difference. Exam tip: any form \(pn+q\) represents an AP.

In a square-number sequence, the nth term is \(n^2\). Therefore, the eighth term is \(8^2=64\). \(49=7^2\) is the seventh term, while \(81=9^2\) is the ninth term. Exam tip: square the term number to find a term in this sequence.

For the fifth term, substitute n = 5. Thus, a₅ = 5² + 3 = 25 + 3 = 28, so 28 is the correct option. The value 27 would result from using 5² + 2, which is not the given rule. Exam tip: while finding an nth term, first substitute the correct value of n and then evaluate the power.

The governing concept is the general term of an arithmetic progression. The first term is a = 2, and the common difference is d = 6 − 2 = 4. Applying a_n = a + (n − 1)d gives a_n = 2 + 4(n − 1) = 2 + 4n − 4 = 4n − 2. Thus option A is correct. Verification is simple: n = 1 gives 4 − 2 = 2, n = 2 gives 8 − 2 = 6, and n = 3 gives 12 − 2 = 10. Option B has the wrong constant, while C and D do not produce the required first term and common difference.

The governing concept is finding the position of a term from an arithmetic sequence's general rule. The sequence starts at 8 and increases by 8 each time, so its nth term is a_n = 8 + (n − 1)8 = 8n. To locate 72, set 8n = 72 and divide both sides by 8: n = 9. Therefore, option C is correct, because 72 is the ninth term. Direct counting also confirms this: the terms are 8, 16, 24, 32, 40, 48, 56, 64 and 72. Options A, B and D correspond to 56, 64 and 80 respectively, not 72.