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**Hint:**Since we are given the sum to infinity is S and we know the formula of sum to infinity is $\dfrac{a}{{1 - r}}$, Where a is the first term term and r is the common ratio and equating both we get the value of r and now using a and r in the formula of sum to n terms, that is ${S_n} = a\left( {\dfrac{{1 - {r^n}}}{{1 - r}}} \right)$we get the required answer.

**Complete step by step solution:**

We are given that the first term of the GP is a

And the sum to infinity is S

In a geometric progression the sum to infinity is given by the formula

Sum to infinity $ = \dfrac{a}{{1 - r}}$

Where a is the first term term and r is the common ratio

Hence we are given the sum to infinity is S

$ \Rightarrow S = \dfrac{a}{{1 - r}}$

From this we can get the value of r by rearranging the terms

$

\Rightarrow S\left( {1 - r} \right) = a \\

\Rightarrow 1 - r = \dfrac{a}{S} \\

\Rightarrow 1 - \dfrac{a}{S} = r \\

$

So now we have the first term to be a and the common ratio is $1 - \dfrac{a}{S} = r$

The Sum to n terms in a GP is given by the formula

$ \Rightarrow {S_n} = a\left( {\dfrac{{1 - {r^n}}}{{1 - r}}} \right)$

Substituting the known values we get

$

\Rightarrow {S_n} = a\left( {\dfrac{{1 - {{\left( {1 - \dfrac{a}{S}} \right)}^n}}}{{1 - \left( {1 - \dfrac{a}{S}} \right)}}} \right) \\

\Rightarrow {S_n} = a\left( {\dfrac{{1 - {{\left( {1 - \dfrac{a}{S}} \right)}^n}}}{{1 - \left( {\dfrac{{S - a}}{S}} \right)}}} \right) \\

\Rightarrow {S_n} = a\left( {\dfrac{{1 - {{\left( {1 - \dfrac{a}{S}} \right)}^n}}}{{\dfrac{{S - S + a}}{S}}}} \right) \\

\Rightarrow {S_n} = a\left( {\dfrac{{1 - {{\left( {1 - \dfrac{a}{S}} \right)}^n}}}{{\dfrac{a}{S}}}} \right) \\

\Rightarrow {S_n} = S\left( {1 - {{\left( {1 - \dfrac{a}{S}} \right)}^n}} \right) \\

$

Hence now we obtained the sum of first n terms

**Therefore the correct answer is option B.**

**Note :**

If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the sequence is called a geometric progression. (GP), whereas the constant value is called the common ratio.

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