Explicit or general rule

The numerator is (n^2+1) and the denominator is (n+2), so (a_n=\frac{n^2+1}{n+2}). In a fractional sequence identify the numerator pattern separately.

The numerator is (n) and the denominator is (2n+1), so (a_n=\frac{n}{2n+1}). In fractions observe the denominator growth carefully.

The governing concept is substitution into an explicit sequence rule, followed by subtraction of two terms. For the eighth term, substitute n = 8: a₈ = 9(8) − 4 = 72 − 4 = 68. For the third term, substitute n = 3: a₃ = 9(3) − 4 = 27 − 4 = 23. Therefore a₈ − a₃ = 68 − 23 = 45, so option B is correct. A useful shortcut is that the constant term cancels: (9 × 8 − 4) − (9 × 3 − 4) = 9(8 − 3) = 45. Option A could result from an arithmetic error, while options C and D do not follow from the required evaluations. Both terms must be calculated using their correct indices before finding the difference.

The correct rule is \(a_n=3n^2+2\). Substituting \(n=1,2,3,4\) gives \(5,14,29,50\), respectively. The closest distractor, \(a_n=3n^2+n\), gives \(4\) when \(n=1\), so it cannot represent the sequence. Exam tip: verify a proposed general rule using at least the first two or three terms.

Given \(a_n=n^2+cn\), substitute \(n=3\): \(a_3=3^2+3c=9+3c\). Since \(a_3=18\), \(9+3c=18\), so \(3c=9\) and \(c=3\). Therefore, option 3 is correct. If \(c=2\), then \(a_3=9+6=15\), not 18. Exam tip: substitute the specified value of \(n\) directly into the general term.

Given \(a_n=12-2n\), for the first negative term we need \(12-2n<0\). This gives \(n>6\), so the smallest integer value of \(n\) is \(7\). Indeed, \(a_6=0\), which is not negative, whereas \(a_7=12-14=-2\). Therefore, the 7th term is the first negative term. Exam tip: a negative term must be less than \(0\); zero is neither positive nor negative.

The increase over three gaps is (12), so (d=4), hence (a_9=25+3(4)=37). Extend the terms using the common difference.

This is (1^3+1,2^3+1,3^3+1,4^3+1,\ldots), so (a_n=n^3+1). In cube sequences identify the constant shift.

Given \(\frac{n(n+1)}{2}=45\), we get \(n(n+1)=90\), so \(n^2+n-90=0\). Factoring gives \((n-9)(n+10)=0\). Since a term number must be positive, \(n=9\). Therefore, 45 is the 9th term of the sequence. The 10th term is \(\frac{10\times11}{2}=55\), so it is not correct. Exam tip: Clear the denominator first and take the positive root of the resulting quadratic equation.

The magnitude is (6,9,12,15,\ldots) and signs alternate, so (a_n=(-1)^{n+1}3(n+1)). Use the first term sign to decide the power of ((-1)).

For an arithmetic sequence, \(a_n=a_1+(n-1)d\). Here, \(a_4=a_1+3(5)=18\), so \(a_1=3\). Hence \(a_n=3+(n-1)5=5n-2\), making option A correct. In option B, \(a_4=22\), so it cannot be correct. Exam tip: when a term \(a_k\) is given, first find \(a_1=a_k-(k-1)d\), then form the general term.

Given \(a_n=2n^2+n-1\), \(a_6=2(6)^2+6-1=72+6-1=77\) and \(a_2=2(2)^2+2-1=8+2-1=9\). Therefore, \(a_6-a_2=77-9=68\). The value 72 is only \(2(6)^2\), so it ignores the \(+6-1\) part of the rule. Exam tip: calculate each required term separately before finding their difference.

The governing concept is finding an explicit rule for an arithmetic sequence. Consecutive terms increase by a constant difference: 13 − 8 = 5, 18 − 13 = 5, and 23 − 18 = 5. An arithmetic sequence has aₙ = a₁ + (n − 1)d, so here aₙ = 8 + (n − 1)5 = 8 + 5n − 5 = 5n + 3. Thus option B is correct. Verification is important: at n = 1, the formula gives 5(1) + 3 = 8; at n = 2 it gives 13; at n = 3 it gives 18; and at n = 4 it gives 23. Option A gives 8, 16, 24, ... and has the wrong difference. Option C begins with 2, and option D has difference 3, so neither describes the given sequence.

\(a_n=5n-2\) has the form \(pn+q\), so \(a_{n+1}-a_n=5\) is constant. For \(n^2+1\), successive differences change. Exam tip: check whether the common difference is constant.

The difference is (9) and the first term is (14) so (a_n=9n+5). In exams take the difference as the coefficient of (n).

At (n=1) it gives (70) and at (n=2) it gives (64) so (a_n=76-6n). In exams match the first term in a decreasing sequence.

The governing concept is finding the position of a term from an explicit formula. Set the given rule equal to the target value: 4n + 9 = 73. Subtract 9 from both sides to obtain 4n = 64, then divide by 4 to get n = 16. Therefore 73 is the sixteenth term, and option C is correct. Direct checking gives a₁₆ = 4(16) + 9 = 64 + 9 = 73. The other options produce different values: a₁₄ = 65, a₁₅ = 69, and a₁₇ = 77. A common error is to subtract 9 correctly but divide incorrectly, or to confuse the value of a term with its position. Here n is the position, while 73 is the term value.

Its rule is (a_n=7n-2) and (7n-2=61) gives (n=9). In exams form the general term first to find the term number.

These terms are (4^2,5^2,6^2,7^2) so (a_n=(n+3)^2). In exams recognize shifted square numbers.

The general term is \(a_n=(n+3)^2\). Substituting \(n=5\), we get \(a_5=(5+3)^2=8^2=64\). Hence, 64 is the correct answer. The value 81 would result only if the quantity inside the bracket were 9. Exam tip: evaluate the expression inside brackets before squaring it.