# The common ratio of an infinite geometric series is 11/16, and its sum is 76 4/5, how do you find the first four terms of the series?

Aug 23, 2017

The first four terms of the series are:

$24 , \frac{33}{2} , \frac{363}{32} , \frac{3993}{512}$

#### Explanation:

The general term of a geometric series is given by the formula:

${a}_{n} = a {r}^{n - 1}$

where $a$ is the initial term and $r$ the common ratio.

The sum of the first $N$ terms of such a series is:

${s}_{N} = \frac{a \left(1 - {r}^{N}\right)}{1 - r}$

If $\left\mid r \right\mid < 1$ then ${\lim}_{N \to \infty} {r}^{N} = 0$ and the sum of the whole series is:

${s}_{\infty} = {\lim}_{N \to \infty} {s}_{N} = \frac{a}{1 - r}$

In our example, we are told:

$\left\{\begin{matrix}r = \frac{11}{16} \\ {s}_{\infty} = 76 \frac{4}{5} = \frac{384}{5}\end{matrix}\right.$

Hence:

$\frac{384}{5} = {s}_{\infty} = \frac{a}{1 - r} = \frac{a}{1 - \frac{11}{16}} = \frac{16 a}{5}$

Multiplying both ends by $\frac{5}{16}$ we find:

$a = 24$

So the first four terms are:

$24$

$24 \cdot \frac{11}{16} = \frac{33}{2}$

$\frac{33}{2} \cdot \frac{11}{16} = \frac{363}{32}$

$\frac{363}{32} \cdot \frac{11}{16} = \frac{3993}{512}$