Explicit or general rule

The difference is (8), and at (n=1) the value must be (5), so (a_n=8n-3). In exams, find the constant using the first term.

Given \(a_n=\frac{n}{2}\). For the sixth term, substitute \(n=6\): \(a_6=\frac{6}{2}=3\). Therefore, the correct answer is 3. Choosing \(6\) would ignore the division by 2 in the formula. Exam tip: to find a particular term, substitute its subscript into the general rule.

This is an arithmetic progression with first term \(\frac{1}{2}\) and common difference \(\frac{1}{2}\). Therefore, \(a_n=\frac{1}{2}+(n-1)\frac{1}{2}=\frac{n}{2}\). Check: for \(n=1\), \(a_1=\frac{1}{2}\), and for \(n=2\), \(a_2=1\). The rule \(a_n=n\) gives the first term as 1, so it is incorrect. Exam tip: substitute the first two or three values of \(n\) to verify a general term.

Substituting \(n=1,2,3,4\) in \(2n^2\) gives \(2,8,18,32\), respectively. Hence, the correct general term is \(a_n=2n^2\). The close distractor \(a_n=4n\) gives 8 for the second term, but it gives 4 rather than 2 when \(n=1\). Exam tip: verify a proposed general term using at least the first three terms.

From (3n-2=13), we get (n=5). In exams, when term number is asked, equate the formula to the given value.

Its rule is (a_n=3n-2), and (3n-2=16) gives (n=6). In exams, form the general term first to find the term number.

The general term is \(a_n=n+4\). Substituting \(n=7\), we get \(a_7=7+4=11\). Therefore, 11 is the correct option. The value 10 would result if 3 were added instead of 4. Exam tip: To find a particular term, substitute its term number for \(n\) in the general rule.

When (n=1), the value should be (5), so (a_n=n+4). In exams, relate (n) to the first term in consecutive-number sequences.

\(a_n=2n-1\) generates the odd natural numbers. Checking \(n=1\) gives \(a_1=1\), and each next term increases by 2. \(2n+1\) would start with 3 instead. In exams, verify a rule using the first term.

This is an arithmetic sequence with first term 12 and common difference 12. For \(n=1\), \(a_n=12n=12\); for \(n=2\), it gives 24; and for \(n=3\), it gives 36. Hence, the correct explicit rule is \(a_n=12n\). The rule \(a_n=24n\) would give 24 as the first term, so it is incorrect. Exam tip: substitute \(n=1\) and \(n=2\) to check an explicit rule.

In option A, \(a_n\) is written only in terms of n, so any term can be found directly. B, C and D require a previous term, so they are recursive rules. Exam tip: a rule involving only n is an explicit rule.

The sequence has a constant difference, so the arithmetic-sequence formula applies. The common difference is d = 14 − 9 = 5. Therefore a_n = a_1 + (n − 1)d = 9 + (n − 1)5 = 9 + 5n − 5 = 5n + 4. Thus option A is correct. Testing the first positions gives a_1 = 5 + 4 = 9, a_2 = 10 + 4 = 14, a_3 = 15 + 4 = 19, and a_4 = 20 + 4 = 24. Option B gives 9, 18, 27, …; option C gives 1 as its first term; and option D gives 9, 10, 11, …. The constant part must be 4, not 0, −4, or 8, in order to fit the first term and the difference.

Substitute \(n=4\): \(a_4=\frac{4(4+1)}{2}=\frac{4\times5}{2}=10\). Therefore, the correct answer is 10. Getting 12 usually results from an error in addition or multiplication. Exam tip: for a general term, substitute the given subscript for \(n\) and simplify step by step.

These are triangular numbers: 1 = 1·2/2, 3 = 2·3/2, 6 = 3·4/2, and 10 = 4·5/2. The nth triangular number is therefore a_n = n(n + 1)/2, so option C is correct. The successive differences are 2, 3, and 4, which also show why a simple arithmetic formula with a constant difference is not appropriate. Substitution gives the listed values for n = 1, 2, 3, and 4. Option A gives 1, 4, 9, 16; option B gives 1, 3, 5, 7; and option D gives 3, 4, 5, 6. Thus the triangular-number formula is the only option that reproduces the complete sequence.

Given \(a_n=100-5n\), substitute \(n=6\): \(a_6=100-5(6)=100-30=70\). Therefore, 70 is the correct option. The value 65 may result from subtracting 5 only once, but here \(5\times6\) must be subtracted. Exam tip: To find a term from an explicit rule, substitute the given value of \(n\) and perform multiplication first.

These terms are (4) times square numbers, so (a_n=4n^2). In exams, substitute (n=1,2,3) to match.

Given \(a_n=2^n-1\), substitute \(n=4\): \(a_4=2^4-1=16-1=15\). Therefore, 15 is the correct option. The closest distractor, 16, is only the value of \(2^4\); the subtraction of 1 must still be done. Exam tip: substitute the term number first, then follow the order of operations.

The first term is (20) and the difference is (-3), so (a_n=23-3n). In exams, treat the common difference as negative in decreasing sequences.

The sequence is formed by consecutive multiples of 4. The first term is 4 × 1, the second is 4 × 2, the third is 4 × 3, and the fourth is 4 × 4. Therefore the explicit general term is a_n = 4n, which makes option A correct. Using the arithmetic formula gives the same result: a_1 = 4 and d = 4, so a_n = 4 + (n − 1)4 = 4n. Substitution confirms that n = 1, 2, 3, and 4 produce 4, 8, 12, and 16. Option B has common difference 1, option C gives 4, 7, 10, …, and option D gives 3 as its first term. Hence only option A matches both the starting value and the constant difference.

In \(a_n=5n-3\), the difference \(a_{n+1}-a_n=5\) is constant, so it is an arithmetic progression. A geometric progression requires a constant ratio, not a constant difference. Exam tip: test consecutive-term differences first.