Question

If the first 5 terms of an infinite geometric sequence are 100, 60,36, 21.6 , and 12.96, then the sum of all the terms in the sequence is _______.

Answer 1

Solution:

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 3/5 gives the next term. In other words,

$a_n\text{ = }a_1r^{n-1}$

where r = 3/5. Now, The sum of a series S_n is calculated using the formula:

$S_n\text{ = }\frac{a(1-r^n)}{1-r}$

For the sum of an infinite geometric series, as n approaches infinity we have that 1-r^n approaches 1. Thus

$\text{ }\frac{a(1-r^n)}{1-r}$

approaches

$\text{ }\frac{a}{1-r}$

then the sum of an infinite geometric series would be:

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

The values a = 100 and r = 3/5 can be put in the previous equation:

$S_{\infty}=\frac{100}{1-\frac{3}{5}}_{}=\text{ }250$

then, the correct answer is:

$S_{\infty}=250$