Explicit or general rule

The difference is (7), and at (n=1) the value must be (12), so (a_n=7n+5). In exams, take the difference as the coefficient of (n).

Given \(a_n=50-4n\), substitute \(n=9\): \(a_9=50-4(9)=50-36=14\). Therefore, 14 is correct. A value such as 16 can result from incorrectly evaluating \(4\times9\). Exam tip: substitute the term number first, then perform multiplication before subtraction.

At (n=1) it gives (42), and at (n=2) it gives (37), so (a_n=47-5n). In exams, always match the first term in a decreasing sequence.

From (3n+2=38), we get (n=12). In exams, equate the formula to the given value when term number is asked.

Its rule is (a_n=3n+2), and (3n+2=32) gives (n=10). In exams, form the general term first to find the term number.

These terms are (2^2,3^2,4^2,5^2), so (a_n=(n+1)^2). In exams, recognize shifted square numbers.

Given (a_n=(n+1)^2), substitute n=6: (a_6=(6+1)^2=7^2=49). Option 36 results from squaring 6 without first adding 1, so it does not follow the given rule. Exam tip: substitute the term number into the rule, simplify the brackets first, and then apply the exponent.

(n(n+1)) gives (2,6,12,20). In exams, check rules involving the product of consecutive numbers.

\(a_{n+1}-a_n=[3(n+1)-2]-(3n-2)=3\), so consecutive terms increase by a constant 3. Option C describes a multiplicative change, which does not apply here. Exam tip: find the common difference first.

Adding (1) to triangular numbers gives (2,4,7,11). In exams, think of triangular numbers when differences are (2,3,4).

The governing idea is direct substitution into an explicit formula. Since the question asks for a₅, replace n by 5 throughout the rule: a₅ = 5(5 + 1)/2 + 1 = 5 × 6/2 + 1. First calculate the product, 5 × 6 = 30; then divide by 2 to obtain 15; finally add the separate constant 1, giving 16. Therefore, option B is correct. A common error is to stop at 15 and forget the final +1, which leads to option A. Options C and D can result from adding too much or mishandling the fraction. The index n identifies the term, while the formula determines its value, so no pattern guess is needed.

Substituting \(n=1,2,3,4\) into \(a_n=n^2-1\) gives \(0,3,8,15\), respectively. Hence, \(a_n=n^2-1\) is the correct general term. For \(a_n=n^2+1\), the first term is \(2\), so it cannot represent the sequence. Exam tip: verify a proposed general term by checking at least \(n=1\) and \(n=2\).

To find the term number, set the explicit rule equal to the required value: n² − 1 = 35. Add 1 to both sides, giving n² = 36. Since a term index is a positive counting number, take n = √36 = 6, not the negative solution −6. Substituting back confirms the result: a₆ = 6² − 1 = 36 − 1 = 35. Therefore, option D is correct. Option A gives 4² − 1 = 15, option B gives 5² − 1 = 24, and option C gives 7² − 1 = 48. This illustrates that finding which term has a given value requires solving the explicit equation and then selecting a valid positive index.

Substituting \(n=1,2,3,4\) into \(a_n=2^n+1\) gives \(3,5,9,17\), respectively. Hence, the correct rule is \(a_n=2^n+1\). The closest distractor, \(2^n-1\), gives \(1\) as the first term, so it is incorrect. Exam tip: verify an explicit rule by substituting values of \(n\) for at least the first three terms.

Given (a_n=2^n+1), substitute n=5: (a_5=2^5+1=32+1=33). The value 31 would result from using (2^5-1), but the rule requires adding 1. Exam tip: substitute the term number first, then evaluate the exponent and remaining operations in order.

For \(a_n=3^n+2\), we get \(a_1=3+2=5\), \(a_2=9+2=11\), \(a_3=27+2=29\), and \(a_4=81+2=83\). Hence, the correct general term is \(a_n=3^n+2\). In contrast, \(a_n=3n+2\) grows linearly and gives \(11\) as its third term, not \(29\). Exam tip: test a proposed rule by substituting \(n=1,2,3\).

Given \(a_n=3^n+2\), substitute \(n=3\): \(a_3=3^3+2=27+2=29\). Hence, 29 is correct. The option 27 is only the value of \(3^3\); the added 2 has been omitted. Exam tip: substitute the term number first, then complete each operation in order.

The sequence is geometric because each term is obtained by multiplying the previous term by 3: 18/6 = 3, 54/18 = 3, and 162/54 = 3. For a geometric sequence, the nth term is aₙ = a₁rⁿ⁻¹. Here a₁ = 6 and r = 3, so aₙ = 6·3ⁿ⁻¹ = 2·3ⁿ, which is option C. Direct checks give 2·3¹ = 6, 2·3² = 18, 2·3³ = 54, and 2·3⁴ = 162. Option A is linear, option B gives 6, 12, 30, 84 rather than the sequence, and option D uses a factor of 2 instead of the required ratio 3. Thus C is the only valid explicit rule.

Substitute \(n=5\) in \(a_n=5\cdot2^{n-1}\): \(a_5=5\cdot2^{5-1}=5\cdot2^4=5\cdot16=80\). Therefore, 80 is correct. Getting 40 may result from incorrectly using \(2^3\). Exam tip: first substitute the term number into the exponent \(n-1\), then calculate.

The first term is (5), and each term is multiplied by (2). In exams, check both the first term and ratio in a geometric sequence.