Question

sum of infinite series is of real numbers is 14.the sum of cubes of the term of the series is 392.what is the first term of the series

Answer 1

common ratio is 2 first term is -14
and common ratio 1/2 first term 7

Answer 2

Given :

The sum of an infinite geometric series of real numbers is 14, and the sum of the cubes of the terms of this series is 392.

To Find:

The first term of the series is

Solution:

We are given that The sum of an infinite geometric series of real numbers is 14.

Formula of sum of infinite terms of GP :

$S_{\infty}=\frac{a}{1-r}$

So, $\frac{a}{1-r}=14 ------1$

We are also given that the sum of the cubes of the terms of this series is 392.

So, $\frac{a^3}{1-r^3}=392 ---- 2$

Cubing equation 1 and Divide 1 and 2

So,$\frac{14^3}{392}=\frac{(\frac{a}{1-r})^3}{\frac{a^3}{1-r^3}}$

$\frac{14^3}{392}=\frac{1-r^3}{(1-r)^3}\\7(1-r)^2=1+r^2-r\\6r^2-13r+6=0\\r=\frac{2}{3},\frac{3}{2}$

For$r =\frac{2}{3}$

$\frac{a}{1-\frac{2}{3}}=14\\a=7$

For$r = \frac{3}{2}$

$\frac{a}{1-\frac{3}{2}}=14$

a=-7

So,the first term of the series is 7 or -7✅