Question

How do you find the sum of the infinite geometric series 12+4+4/3+...?

Answer 1

`18`

Explanation:

An infinite geometric series of the form `sum_(n=1)^ooar^(n-1)` converges if and only if `|r|<1`, where r is the common ration between terms and given by `r=x_(n+1)/(x_n)`.
In this case it converges to the value `a/(1-r)` where a is the first term in the corresponding sequence `(x_n)`.

So in this case, `r=4/12=4/3/4=1/3<1` hence the series converges.

Its sum is hence `sum_(n=1)^oo12*(1/3)^(n-1)=12/(1-1/3)=18`