A lvl H2 Maths: APGP

AJC Prelims 2000

The n^th term of an infinite geometric series is equal to one fourth of the sum of all the terms after ( but not including ) the n^th term. Show that the sum to infinity of the geometric series is five times its first term.

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Answer:

T_n = 1/4 (S_∞ - S_n)
$ar^{n-1}=\frac{1}{4}\left ( \frac{a}{1-r}-\frac{a\left [ 1-r^{n} \right ]}{1-r} \right )$
$4ar^{n-1}=\frac{a}{1-r}\left ( 1-\left ( 1-r^{n} \right ) \right )$
$4ar^{n-1}=\frac{a}{1-r}\left ( r^{n} \right )$
$4=\frac{r}{1-r}$
4 - 4r = r
5r = 4
r = 4/5

Hence, S_∞ = a / (1 - r) = 5a (shown)

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A lvl H2 Maths: APGP