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# The sequence S is defined by Sn – 1 = (Sn) for each integer

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The sequence S is defined by Sn – 1 = (Sn) for each integer [#permalink]  25 Jul 2018, 17:31
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Question Stats:

74% (01:08) correct 25% (01:08) wrong based on 62 sessions
The sequence S is defined by $$S_{n – 1} = \frac{1}{4}(S_n)$$ for each integer n ≥ 2. If $$S_1 = –4$$, what is the value of $$S_4$$?
(A) –256
(B) –64
(C)$$\frac{-1}{16}$$
(D)$$\frac{1}{16}$$
(E) 256
[Reveal] Spoiler: OA

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Sandy
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Retired Moderator
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2934 [1] , given: 394

Re: The sequence S is defined by Sn – 1 = (Sn) for each integer [#permalink]  12 Aug 2018, 04:42
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Explanation

The sequence $$S_{n – 1} = \frac{1}{4}(S_n)$$ can be read as “to get any term in sequence S, multiply the term after that term by $$\frac{1}{4}$$.” Since this formula is “backwards” (usually, later terms are defined with regard to previous terms), solve the formula for $$S_n$$:

$$S_{n – 1} = \frac{1}{4}(S_n)$$
$$4S_{n – 1} = S_n$$
$$S_n = 4S_{n – 1}$$

This can be read as “to get any term in sequence S, multiply the previous term by 4.”

The problem gives the first term and asks for the fourth:

 -4 $$S_1$$ $$S_2$$ $$S_3$$ $$S_4$$

To get $$S_2$$, multiply the previous term by 4: $$(4)(-4) = -16$$. Continue this procedure to find each subsequent term. Therefore, $$S_3 = (4)(-16) = -64$$. $$S_4 = (4)(-64) = -256$$:

 -4 -16 -64 -256 $$S_1$$ $$S_2$$ $$S_3$$ $$S_4$$

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Re: The sequence S is defined by Sn – 1 = (Sn) for each integer [#permalink]  10 May 2020, 03:49
A nice little trick here is to multiply both sides by 4 so it all reads well.
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Re: The sequence S is defined by Sn – 1 = (Sn) for each integer [#permalink]  31 May 2020, 18:16
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If you try to mess with the equation as it stands, it can feel kind of confusing. So, "solve" for Sn by multiplying both sides by 4 to get

Sn = 4(Sn-1)

Ah, much better. Since you know that n=1 --> -4, you can plug that in for Sn-1 to get the value for n=2. You'll get -16, for n=3 you'll get -64, and for n=4 you'll get -256.

Alternatively, you can notice that the function aligning to this sequence is f(n)=-4^n. Plug in 4 and you'll get -256.
Re: The sequence S is defined by Sn – 1 = (Sn) for each integer   [#permalink] 31 May 2020, 18:16
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