If S is sum to infinity of a G.P. , whose first term is a , then sum of first n terms is

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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

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