Question

The sum of the first n terms of an infinite G.P.is
A. \[S=\dfrac{a}{1-r}\]
B. \[{{S}_{n}}=\dfrac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}\]
C. \[S=\dfrac{an}{1-r}\]
D. \[S=\dfrac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1+r}\]

Answer 1

Hint: Consider \[a+ar+a{{r}^{2}}+.....+a{{r}^{n}}+......\infty \] be the infinite series. Find the sum of n terms of G.P. and apply the condition -1 < r < 1. Thus take the limit of the sum of n terms in the condition of infinite G.P.

Complete step-by-step answer:
If a sequence is geometric there are ways to find the sum of first n terms denoted as \[{{S}_{n}}\]. To find the sum of the first \[{{S}_{n}}\] terms of a geometric sequence use the formula,
\[{{S}_{n}}=\dfrac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r},r\ne 1\], where n is the number of terms, a is the first term and r is the common ratio.
Now let us consider the series of the form,
\[a+ar+a{{r}^{2}}+.....+a{{r}^{n}}+......\infty \] is called an infinite geometric series.
Let the common ratio, r be -1 < r < 1, i.e. \[\left| r \right|<1\].
Therefore the sum of n terms of G.P. is given by,
\[{{S}_{n}}=a\left( \dfrac{1-{{r}^{n}}}{1-r} \right)=\dfrac{a-a{{r}^{n}}}{1-r}=\dfrac{a}{1-r}-\dfrac{a{{r}^{n}}}{1-r}.......(1)\]
Since -1 < r < 1, therefore \[{{r}^{n}}\] decreases as n increases and \[{{r}^{n}}\] tends to zero as n tends to infinity, i.e. \[{{r}^{n}}\to 0\] and \[n\to \infty \].
Therefore, \[\dfrac{a{{r}^{n}}}{a-r}\to 0\] as \[n\to \infty \].
Hence from equation (1), the sum of infinite G.P. is given by,
\[S=\underset{x\to 0}{\mathop{\lim }}\,{{S}_{n}}=\underset{x\to \infty }{\mathop{\lim }}\,\left( \dfrac{a}{1-r}-\dfrac{a{{r}^{2}}}{1-r} \right)=\dfrac{a}{1-r}\], if \[\left| r \right|<1\]
Thus the sum of first n terms of an infinite \[G.P.=S=\dfrac{a}{1-r}\].
Option A is the correct answer.

Note: The infinite G.P. series we have considered, \[a+ar+a{{r}^{2}}+.....+a{{r}^{n}}+......\infty \]
has sum when-1 < r < 1. Thus we can say that the given series is convergent.
If an infinite series has a sum, the series is said to be convergent. On the contrary, an infinite series is said to be divergent, it has no sum.