Explicit or general rule

Correct answer: C. 9

Explanation: The general rule gives each term by substituting its position for \(n\). Putting \(n=2\), we get \(a_2=3^2=9\). Option 8 is incorrect because it equals \(2^3\), whereas the base here is 3 and the exponent is 2. Exam tip: first substitute the term number for \(n\), then evaluate the power.

Correct answer: B. \(a_n=3^n\)

Explanation: This is a geometric sequence in which each term is 3 times the previous term. Starting with \(n=1\), we get \(a_1=3^1=3\), \(a_2=3^2=9\), \(a_3=3^3=27\), and \(a_4=3^4=81\). Hence, the correct general term is \(a_n=3^n\). The rule \(a_n=3n\) would give 3, 6, 9, 12, so it is not correct. Exam tip: Substitute \(n=1\) in a proposed rule to check the first term.

Correct answer: C. \(20\)

Explanation: Given \(a_n=n^2-n\), substitute \(n=5\): \(a_5=5^2-5=25-5=20\). Hence, \(20\) is correct. \(25\) is only the value of \(5^2\); subtracting \(5\) is also required. Exam tip: substitute the value of \(n\) in every occurrence in the formula.

Correct answer: A. \(a_n=n^2-n\)

Explanation: For term numbers \(n=1,2,3,4\), the rule \(n^2-n\) gives \(0,2,6,12\), respectively. Therefore, the correct general term is \(a_n=n^2-n\). The rule \(a_n=2n\) gives 2 as its first term, so it does not match the sequence. Exam tip: verify a general term by substituting \(n=1\) and \(n=2\) and checking the first two terms.

Correct answer: A. Constant sequence

Explanation: Since \(a_n=7\), every term remains 7 regardless of \(n\), so it is a constant sequence. Its common difference is 0, not 7. Exam tip: list terms as 7, 7, 7 to identify it quickly.

Correct answer: A. a_n = 4n + 3

Explanation: The sequence follows an explicit arithmetic rule. Its common difference is 11 − 7 = 4, so the coefficient of n in the general term is 4. Applying a_n = a_1 + (n − 1)d gives a_n = 7 + (n − 1)4 = 7 + 4n − 4 = 4n + 3. Therefore option A is correct. Substitution confirms that n = 1 gives 7, n = 2 gives 11, n = 3 gives 15, and n = 4 gives 19. Option B gives 7, 14, 21, …; option C starts with 1; and option D gives 7, 8, 9, …. These alternatives either use the wrong difference or fail to preserve the pattern, so they cannot be the required general rule.

Correct answer: A. It is an AP with common difference 5

Explanation: In \(a_n=5n+2\), the coefficient of \(n\) is 5, so each successive term increases by 5. Hence it is an AP, not a GP. Exam tip: in the form \(pn+q\), the common difference is \(p\).

Correct answer: B. (a_n=8n-3)

Explanation: 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.

Correct answer: B. 3

Explanation: 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.

Correct answer: B. \(a_n=\frac{n}{2}\)

Explanation: 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.

Correct answer: C. \(a_n=2n^2\)

Explanation: 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.

Correct answer: C. (n=5)

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

Correct answer: C. (6)th term

Explanation: 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.

Correct answer: C. 11

Explanation: 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.

Correct answer: A. (a_n=n+4)

Explanation: 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.

Correct answer: A. \(a_n=2n-1\)

Explanation: \(a_n=2n-1\) 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.

Correct answer: B. \(a_n=12n\)

Explanation: 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.

Correct answer: A. \(a_n=3n-2\)

Explanation: 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.

Correct answer: A. a_n = 5n + 4

Explanation: The relevant concept is the general term of an arithmetic sequence. The consecutive difference is constant: 14 − 9 = 5, 19 − 14 = 5, and 24 − 19 = 5. The arithmetic formula is a_n = a_1 + (n − 1)d. With a_1 = 9 and d = 5, it becomes a_n = 9 + 5(n − 1) = 9 + 5n − 5 = 5n + 4. Thus option A is correct. Checking n = 1, 2, 3, and 4 gives 9, 14, 19, and 24. Option B produces 9, 18, 27, so its difference is 9. Option C starts at 1, and option D produces 9, 10, 11, so its difference is only 1. Only A fits every term.

Correct answer: B. 10

Explanation: 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.

Correct answer: C. a_n = n(n + 1)/2

Explanation: The governing idea is recognition of triangular numbers and their explicit formula. Each term can be expressed as the product of two consecutive positive integers divided by 2: 1 = 1×2/2, 3 = 2×3/2, 6 = 3×4/2, and 10 = 4×5/2. Therefore the term in position n is a_n = n(n + 1)/2, making option C correct. The successive differences are 2, 3, and 4, so this is not an arithmetic sequence with a constant difference. Option A gives square numbers 1, 4, 9, 16. Option B gives odd numbers 1, 3, 5, 7, while option D gives 3, 4, 5, 6. None of these alternatives reproduces all the displayed terms.

Correct answer: C. 70

Explanation: 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.

Correct answer: B. (a_n=4n^2)

Explanation: Direct answer: B, \\(a_n=4n^2\\). The terms are 4, 16, 36 and 64. These can be written as \\(4\times1^2\\), \\(4\times2^2\\), \\(4\times3^2\\) and \\(4\times4^2\\). Therefore the rule is \\(a_n=4n^2\\). Step 1: identify the square numbers 1, 4, 9 and 16. Step 2: multiply each by 4 to obtain 4, 16, 36 and 64. Step 3: replace the position number by n. Option B gives exactly these terms. Option A gives 2, 8 and 18, so it is too small and has the wrong multiplier. Option C gives 5 at \\(n=1\\), not 4, and does not preserve the pattern. Option D gives 4, 8 and 12, which is a linear sequence rather than the given square-based sequence. Exam cue: test a proposed rule at n=1, 2 and 3 before accepting it.

Correct answer: C. 15

Explanation: 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.

Correct answer: A. (a_n=23-3n)

Explanation: 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.