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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] New post 25 Jul 2018, 17:31
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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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Retired Moderator
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Kudos [?]: 2934 [1] , given: 394

Re: The sequence S is defined by Sn – 1 = (Sn) for each integer [#permalink] New post 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] New post 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] New post 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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The sequence S is defined by Sn – 1 = (Sn) for each integer

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