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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A nice little trick here is to multiply both sides by 4 so it all reads well.

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.

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