ExplanationThe 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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