$S$ is a sequence $s_1, s_2, s_3, \ldots, s_n$ in which every term after the first is one less than three times the

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 amFollowed by: 2 members

by Gmat_mission » Wed Jun 24, 2020 8:08 am

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$S$ is a sequence $s_1, s_2, s_3, \ldots, s_n$ in which every term after the first is one less than three times the previous term. If $s_5 - s_3 = 28,$ which of the following is the first term in the sequence?

A. $2/3$
B. $8/9$
C. $1$
D. $5/3$
E. $2$

[spoiler]OA=B[/spoiler]

Source: Princeton Review

Source: Beat The GMAT — Problem Solving | Wed Jun 24, 2020 8:08 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

Re: $S$ is a sequence $s_1, s_2, s_3, \ldots, s_n$ in which every term after the first is one less than three times

by deloitte247 » Fri Jun 26, 2020 12:14 am

$$Let\ s_1=x$$
$$s_2=3x-1$$
$$s_3=3\left(3x-1\right)-1=9x-3-1=9x-4$$
$$s_4=3\left(9x-4\right)-1=27x-12-1=27x-13$$
$$s_5=3\left(27x-13\right)-1=81x-39-1=81x-40$$
$$Given\ that\ \ s_5-s_3=28$$
$$\left(81x-40\right)-\left(9x-4\right)=28$$
$$81x-40-9x+4=28$$
$$81x-9x=28+36$$
$$\frac{72x}{72}=\frac{64}{72}$$
$$x=\frac{64}{72}=\frac{16}{18}=\frac{8}{9}$$
$$Answer\ =\ B$$

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8246; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: $S$ is a sequence $s_1, s_2, s_3, \ldots, s_n$ in which every term after the first is one less than three times

by Scott@TargetTestPrep » Mon Jun 29, 2020 5:54 am

Gmat_mission wrote: ↑
Wed Jun 24, 2020 8:08 am
$S$ is a sequence $s_1, s_2, s_3, \ldots, s_n$ in which every term after the first is one less than three times the previous term. If $s_5 - s_3 = 28,$ which of the following is the first term in the sequence?

A. $2/3$
B. $8/9$
C. $1$
D. $5/3$
E. $2$

[spoiler]OA=B[/spoiler]

Solution:

Let can let s1 = x, so in terms of x, we see that

s2 = 3x - 1,

s3 = 3(3x - 1) - 1 = 9x - 4,

s4 = 3(9x - 4) - 1 = 27x - 13, and

s5 = 3(27x - 13) - 1 = 81x - 40.

Since s5 - s3 = 28, in terms of x, we have:

81x - 40 - (9x - 4) = 28

72x - 36 = 28

72x = 64

x = 64/72 = 8/9

Answer: B

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