Question

The sum of an infinite geometric series is 27 times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio?

Geometric Series
Geometric Sequence Formula:
a sub n= a sub 1 `*` `r^(n-1)`
where:
a sub n= the term you are looking for
a sub 1= the first term (sorry I don't know how to format these)
r= common ratio
n= the number of the term you are looking for (ex: if looking for 8th term, n is 8)

Answer 1

Common ratio is `1/3`

Explanation:

Let the first term be `a` and common ratio is `r`, `n^(th)` term is `ar^((n-1))` i.e. fourth term is `ar^3`. Observe that as we have a limiting infinite series `|r|` < 1#.

Now sum of infinite series is `a/(1-r)`

and if first three term are removed, fourth term becomes the first term i.e. first term becomes `ar^3` and

sum of infinite series becomes `(ar^3)/(1-r)`

As `a/(1-r)=27xx(ar^3)/(1-r)`

`27r^3=1` or `r^3=1/27` and `r=1/3` and hence

Common ratio is `1/3`