Find the ratio of a geometric sequence such that its sum is $4$ times the first term

Question

How to find the sum to infinity: the sum to infinity of a geometric progression is 4 times the first term. Find the common ratio.

Answer 1

The sum to infinity could be calculated only for progression with absolute value of the common ratio less than one $|r|<1$ (cause else the sum will be equal to $\infty$). If $a$ is the first term, then

$$ \sum _{k=0} ^{\infty} a r ^k = \frac{a}{1 -r}$$

but the sum also equals to $4a$, then

$$ 4 a = \frac{a}{1-r} $$

$$ 4 = \frac{1}{1-r} $$

$$ \frac{1}{4} = 1 - r $$

$$ r = \frac{3}{4} $$

Let's check: $r = \frac{3}{4}$

$$ \sum _{k=0} ^{\infty} a r ^k = \frac{a}{1 -\frac{3}{4}} = \frac{a}{\frac{1}{4}} = 4a$$