nth term

The nth term is \(a_n=6n-1\). Substituting \(n=15\), we get \(a_{15}=6\times15-1=90-1=89\). Therefore, 89 is correct. The value 91 would result from ignoring the \(-1\) in the formula. Exam tip: for an nth-term question, substitute the required term number for \(n\) in the given formula.

For the term to be 33, set \(2^n+1=33\). Thus, \(2^n=32\). Since \(32=2^5\), we get \(n=5\), so the 5th term is 33. The 4th term is \(2^4+1=17\), so it is not correct. Exam tip: In exponential-term questions, first move the constant to the other side and express the result as a power of the same base.

Look at the terms by separating a power of 2 from the remaining constant: 5 is \(2^2+1\), 9 is \(2^3+1\), 17 is \(2^4+1\), 33 is \(2^5+1\), and 65 is \(2^6+1\). The exponent starts at 2 when the position is 1, so the exponent in the nth term is \(n+1\). Hence the general term is \(a_n=2^{n+1}+1\).

Checking the formula confirms it: for n=1 it gives \(2^2+1=5\), and for n=5 it gives \(2^6+1=65\). Therefore option A is correct. Option B has the right type of expression but gives 5 for the first term, while option D starts with a different power and gives an incorrect value. Recognizing the doubling pattern and the fixed added 1 is the key step.

(a_9=108) and (a_4=28), so the difference is (88), not (80). Open brackets carefully in product rules.

The sequence is formed by squaring the positive counting numbers: the first term is 1 squared, the second is 2 squared, the third is 3 squared, and so on. Thus its general term is \\(a_n=n^2\\). To find the position of 196, solve \\(n^2=196\\). Since \\(14^2=196\\) and the term number is positive, \\(n=14\\). Therefore, 196 is the 14th term, so option A is correct. The other listed positions would produce different squares.

The pattern can also be checked directly: the terms are 1, 4, 9, 16, 25, ..., which are the squares of 1, 2, 3, 4, 5, ... . Continuing this rule, the number obtained from the 14th position is \\(14^2=196\\). It is not necessary to count every term individually. Taking the positive square root gives the term number because sequence positions are positive integers. The supplied answer A and its reasoning are accurate.

Given \(a_n=12-5n\), \(a_3=12-5(3)=-3\) and \(a_8=12-5(8)=-28\). Therefore, \(a_3+a_8=-3+(-28)=-31\). The value \(-29\) can result from an error while substituting the term number or handling the negative sign. Exam tip: find each required term separately before adding them.

Given \(a_n=\frac{n(n+1)}{2}\). Substituting \(n=12\), \(a_{12}=\frac{12(12+1)}{2}=\frac{12\times13}{2}=78\). The value 72 may result from incorrectly using 12 in place of \(n+1\). Exam tip: substitute the given value of \(n\) carefully before simplifying the formula.

The terms are (1\cdot2,2\cdot3,3\cdot4,\ldots), so (a_n=n(n+1)). Recognize products of consecutive numbers.

Given (a_n=3^n-2), substitute n=4: (a_4=3^4-2=81-2=79). Therefore, 79 is correct. The closest distractor, 81, is only the value of (3^4); 2 must still be subtracted. Exam tip: for an nth-term formula, substitute the given value of n first, evaluate powers, and then perform the remaining operations.

For odd (n), terms are negative and for even (n), terms are positive, so (a_n=(-1)^n n). Check the sign pattern separately.

To find the first term, put n=1. Thus, a_1=2(1)+9=11, so 11 is correct. The number 9 is only the constant term, not the first term, because the contribution of 2n must also be included when n=1. Exam tip: substitute n=1 in any nth-term formula to obtain the first term.

Given \(a_n=4n^2-1\), substitute \(n=5\) to find the fifth term: \(a_5=4(5)^2-1=4\times25-1=99\). Therefore, 99 is correct. The value 101 would result from incorrectly using \(+1\) instead of the final \(-1\). Exam tip: substitute the term number first and evaluate the exponent before other operations.

The first term is (11) and the difference is (7), so (a_{25}=11+24\cdot7=179). The (25)th term includes (24) differences.

Direct answer: Option A, 25. The formula is \\(a_n=2n-1\\). Substitute n=1,2,3,4,5 one at a time: \\(a_1=2(1)-1=1\\), \\(a_2=2(2)-1=3\\), \\(a_3=5\\), \\(a_4=7\\), and \\(a_5=9\\). Now add them: \\(1+3+5+7+9=25\\). Option A is correct. Option B, 20, is too small and does not equal the calculated sum. Option C, 30, is too large; it may result from adding incorrectly or including an extra amount. Option D, 15, is also incorrect and is not the sum of these five terms. The terms form the first five positive odd numbers, and their sum is 25. A useful alternative check is the formula for the first n odd numbers, \\(1+3+\cdots+(2n-1)=n^2\\); for n=5, the sum is \\(5^2=25\\). Memory cue: when only five simple terms are required, write them first and add carefully.

Given \(a_n=10n+4\), substitute \(n=r+2\): \(a_{r+2}=10(r+2)+4=10r+20+4=10r+24\). Hence, \(10r+24\) is correct. In \(10r+14\), the product \(10\times2=20\) has been handled incorrectly. Exam tip: always substitute the complete index expression in brackets.

(a_2=2) and (a_5=11), so the sum is (13). The listed options do not contain the correct value.

Given \(a_n=2n^2+n\). Substituting \(n=6\), \(a_6=2(6)^2+6=2\times36+6=78\). The option 72 can result from forgetting to add the linear term \(+6\). Exam tip: substitute the value of \(n\) first, then evaluate the exponent carefully.

This is an arithmetic sequence because the common difference between consecutive terms is \(-4\). Here, the first term is \(a=13\) and \(d=-4\). Therefore, \(a_n=a+(n-1)d=13+(n-1)(-4)=17-4n\). Hence, \(17-4n\) is correct. In \(13-4n\), substituting \(n=1\) gives 9 instead of the first term 13. Exam tip: Always substitute \(n=1\) to check whether a proposed nth-term formula gives the first term.

The second differences are (2), and the terms match (n^2+3n). Equal second differences suggest a quadratic rule.

Given (a_n=n^3-n), substitute n=5 to find the fifth term: (a_5=5^3-5=125-5=120). Hence, the correct answer is 120. The value 125 is only (5^3); the subtraction of 5 must also be included. Exam tip: while finding an nth term, substitute the value of n in every part of the formula.

This is a geometric sequence because each term is obtained by multiplying the previous term by 2. Here, the first term is \(a=7\) and the common ratio is \(r=2\). Therefore, \(a_n=ar^{n-1}\). So, \(a_{10}=7\times2^9=7\times512=3584\). The option 1792 uses \(2^8\), which gives the 9th term. Exam tip: In a geometric sequence, the exponent for the \(n\)th term is always \(n-1\).

Putting n=4, a_4=6(4)^2-6(4)+1=6×16-24+1=96-24+1=73. Therefore, the correct answer is 73. The value 72 may result from omitting the final +1, which is incorrect. Exam tip: Substitute the value of n in every term first, then simplify in order.

Observe that \(2=1^2+1\), \(5=2^2+1\), \(10=3^2+1\), \(17=4^2+1\), and \(26=5^2+1\). Therefore, the \(n\)th term is \(a_n=n^2+1\). The close distractor \(n^2+2\) gives 3 as its first term, so it does not fit. Exam tip: match the terms with squares of \(1,2,3,\ldots\) to identify a quadratic pattern quickly.

For an arithmetic sequence, the nth term is found from \\(a_n=a_1+(n-1)d\\), where \\(a_1\\) is the first term and \\(d\\) is the common difference. We are given \\(a_6=31\\) and \\(d=4\\). Substituting the sixth position gives \\(31=a_1+5(4)\\), so \\(a_1=31-20=11\\). Therefore, \\(a_n=11+(n-1)4=11+4n-4=4n+7\\). Hence option A is correct.

The check at the given position confirms the result: putting \\(n=6\\) into \\(4n+7\\) gives \\(4(6)+7=31\\), exactly as required. The expression \\(31+4n\\) incorrectly treats 31 as the first term and gives 55 at the sixth position. The other linear options also fail to preserve the stated sixth term or common difference. Thus the supplied answer A is mathematically consistent and the reasoning uses the correct relation for an arithmetic progression.

The term numbers differ by (2) and the common difference is (8), so the difference is (16). Use (d) in linear terms to save time.