Explicit or general rule

An explicit rule gives the value of any term directly from its position n. In this sequence, the first term is 11, the second is 22, the third is 33, and the fourth is 44. Each term is 11 times its position: 11 × 1 = 11, 11 × 2 = 22, 11 × 3 = 33, and 11 × 4 = 44. Therefore the general rule is aₙ = 11n, so option C is correct. Option A gives 11 for n = 1 but then gives 21 for n = 2, while option B gives 12 for the first term. Option D gives 22 as the first term, so it does not match the sequence.

\(a_n=3n-2\) has the linear form \(pn+q\). Here, \(a_{n+1}-a_n=3\), which is constant for every term, so it is an arithmetic progression. For \(n^2+1\), the differences are not constant. Exam tip: test consecutive-term differences.

The sequence increases by a constant difference of 5, so it is an arithmetic sequence. For an arithmetic sequence, the nth term is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. Here a₁ = 3 and d = 5, so aₙ = 3 + (n − 1)5 = 3 + 5n − 5 = 5n − 2. Thus option A is correct. Checking confirms the result: for n = 1, 5(1) − 2 = 3; for n = 2, the value is 8; for n = 3, it is 13; and for n = 4, it is 18. The other formulas fail at one or more of these positions.

Given \(a_n=n^2+3\), substitute \(n=5\) to find the fifth term: \(a_5=5^2+3=25+3=28\). Therefore, the correct answer is \(28\). \(25\) is only the value of \(5^2\); the \(+3\) must also be added. Exam tip: Substitute the term number first, then evaluate the exponent before completing the calculation.

For \(a_n=n^2+3\), substituting \(n=1,2,3,4\) gives \(4,7,12,19\), respectively. Hence, \(a_n=n^2+3\) is the correct general term. Although \(a_n=3n+1\) gives the first two terms 4 and 7, its third term is 10, not 12. Exam tip: test a proposed general term using at least the first three values of \(n\).

Using the rule \(a_n=14-3n\), substitute \(n=3\): \(a_3=14-3(3)=14-9=5\). Therefore, 5 is correct. Getting 7 would result from subtracting only 3 from 14 instead of calculating \(3\times3\). Exam tip: substitute the term number first, then perform multiplication before subtraction.

At (n=1) it gives (11), and at (n=2) it gives (8), so (a_n=14-3n). In exams, treat a decreasing difference as negative.

The general term is \(a_n=5^n\). Substituting \(n=2\) gives \(a_2=5^2=25\). The nearby distractor \(10\) is incorrect because it is not the square of \(5\). Exam tip: To find a particular term from a general rule, substitute the term number for \(n\).

The first term is 5, and each successive term is multiplied by 5: \(5,5^2,5^3,5^4,\ldots\). Therefore, when counting starts at \(n=1\), the general term is \(a_n=5^n\). The rule \(a_n=5^{n-1}\) gives 1 as the first term, so it is incorrect. Exam tip: substitute \(n=1\) to check a proposed general rule.

It is an arithmetic progression because \(a_{n+1}-a_n=[4(n+1)+7]-(4n+7)=4\), a constant difference. A geometric progression needs a constant ratio, not a difference. Exam tip: in \(a_n=dn+c\), \(d\) is the common difference.

Substituting \(n=1,2,3,4\) in \(a_n=3n^2\) gives \(3,12,27,48\), respectively. Therefore, the correct general term is \(a_n=3n^2\). Although \(a_n=3n\) looks related, it gives \(3,6,9,12\), so it is not correct. Exam tip: verify a proposed general term using at least the first three terms.

Given \(a_n=\frac{3n}{2}\). To find the fourth term, substitute \(n=4\): \(a_4=\frac{3\times4}{2}=\frac{12}{2}=6\). Therefore, the correct answer is 6. A value such as 5 may result from substituting an incorrect value of \(n\). Exam tip: to find a particular term, substitute its term number in the general-term formula.

The terms are consecutive multiples of 3/2. Substituting the position n gives a₁ = (3/2)×1 = 3/2, a₂ = (3/2)×2 = 3, a₃ = (3/2)×3 = 9/2, and a₄ = (3/2)×4 = 6. Therefore the explicit rule is aₙ = 3n/2, so option B is correct. This also follows from the arithmetic-sequence formula: the first term is 3/2 and the common difference is 3/2, giving aₙ = 3/2 + (n − 1)(3/2) = 3n/2. Option A decreases the scale by dividing n by 3, while options C and D produce 2 and 3 for the first term instead of 3/2.

From (2n-3=15), we get (n=9). In exams, equate the formula to the given value when term number is asked.

Its rule is (a_n=2n-3), and (2n-3=15) gives (n=9). In exams, form the general term first to find the term number.

Given \(a_n=n^3\). For the third term, substitute \(n=3\): \(a_3=3^3=3\times3\times3=27\). Option 9 is \(3^2\), so it would result from squaring rather than cubing. Exam tip: in an explicit rule, substitute the term number directly for \(n\).

The governing pattern is the sequence of consecutive perfect cubes. The terms can be written as 1³, 2³, 3³, and 4³: 1³ = 1, 2³ = 8, 3³ = 27, and 4³ = 64. Since the nth term is the cube of the position number, the explicit rule is aₙ = n³, making option B correct. Option A gives the square sequence 1, 4, 9, 16, not the given values. Option C produces 3, 6, 9, 12, and option D produces 2, 4, 8, 16 when n begins at 1. Testing the first few positions is sufficient to distinguish the correct rule clearly.

Substitute 2 for n in the general term: \(a_2=2^3+1=8+1=9\). Therefore, the correct answer is 9. Option 8 is only the value of \(2^3\); it misses the given \(+1\). Exam tip: while finding a term of a sequence, substitute the value of n first and then follow the order of operations.

For \(a_n=n^3+1\), substituting \(n=1,2,3,4\) gives \(2,9,28,65\), respectively. Hence, option B is correct. In option A, the third term would be \(10\), not \(28\). Exam tip: verify a proposed general term by checking at least the first three values of \(n\).

Given \(a_n=20-4n\). To find the fourth term, substitute \(n=4\): \(a_4=20-4(4)=20-16=4\). Therefore, 4 is the correct answer. Option 8 could result from incorrectly subtracting 12 instead of calculating \(4\times4\). Exam tip: In an explicit rule, substitute the term number carefully for \(n\).