The sequence \(a_1, a_2, a_3, a_4, a_5\) is such that \(a_n=a_{n-1}+5\) for \(2 \le n \le 5.\) If \(a_5=31,\) what is

[Vincen]

The sequence \(a_1, a_2, a_3, a_4, a_5\) is such that \(a_n=a_{n-1}+5\) for \(2 \le n \le 5.\) If \(a_5=31,\) what is the value of \(a_1?\)

(A) 1
(B) 6
(C) 11
(D) 16
(E) 21

[spoiler]OA=C[/spoiler]

Source: Official Guide

[deloitte247]

$$Given\ that:\ a_n=a_{n-1}+5\ when\ 2\le n\le5$$
$$if\ n=2$$
$$a_2=a_{2-1}+5$$
$$a_2=a_1+5$$
$$and\ a_1=a_2-5$$
$$so\ the\ formula\ for\ this\ sequence\ can\ be\ rephrased\ as\ a_n=a_{n+1}-5$$
$$a_5=31$$
$$a_4=a_5-5=31-5=26$$
$$a_3=a_4-5=26-5=21$$
$$a_2=a_3-5=21-5=16$$
$$a_1=a_2-5=16-5=11$$
$$Answer\ =\ C$$

[Scott@TargetTestPrep]

The sequence \(a_1, a_2, a_3, a_4, a_5\) is such that \(a_n=a_{n-1}+5\) for \(2 \le n \le 5.\) If \(a_5=31,\) what is the value of \(a_1?\)

(A) 1
(B) 6
(C) 11
(D) 16
(E) 21

[spoiler]OA=C[/spoiler]

Source: Official Guide

Solution:

Manipulating the equation, we have a(n - 1) = a(n) - 5 for 2 ≤ n ≤ 5. Now, let’s work backward:

Since a(5) = 31, then a(4) = a(5) - 5 = 31 - 5 = 26.

Since a(4) = 26, then a(3) = a(4) - 5 = 26 - 5 = 21.

Since a(3) = 21, then a(2) = a(3) - 5 = 21 - 5 = 16.

Since a(2) = 16, then a(1) = a(2) - 5 = 16 - 5 = 11.

Answer: C