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8. Sequences and Series
easy
If $S$ is the sum to infinity of a $G.P.$, whose first term is $a$, then the sum of the first $n$ terms is
A
$S{\left( {1 - \frac{a}{S}} \right)^n}$
B
$S\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]$
C
$a\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]$
D
None of these
Solution
(b) Let $r$ be the common ratio of the $G.P. $ Then
$S = \frac{a}{{1 – r}}$
$ \Rightarrow $$r = 1 – \frac{a}{S}$
Now ${S_n} = $ Sum of $n$ terms
$ = a\left( {\frac{{1 – {r^n}}}{{1 – r}}} \right) = \frac{a}{{1 – r}}(1 – {r^n}) = S\left[ {1 – {{\left( {1 – \frac{a}{S}} \right)}^n}} \right]$.
Standard 11
Mathematics
