Question

How do I find the common ratio of an geometric sequence on a calculator?

Answer 1

Assuming the terms are nonzero, we can find the common ratio `r` on a calculator by taking any two consecutive terms and dividing the later one by the earlier one:

`r= a_(n+1)/a_n`

Explanation:

A geometric sequence is a sequence with a common ratio `r` between adjacent terms, that is, a sequence of the form `a_1, a_1r, a_1r^2, ..., a_1r^n, ...`

Then, assuming the terms are nonzero, dividing any term by the prior term will give the common ratio:

`(cancel(a_1)r^n)/(cancel(a_1)r^(n-1))=r^n/r^(n-1)=r^(n-(n-1))=r^1=r`

To find `r` on a calculator, then, take any two consecutive terms and divide the later one by the earlier one.

In fact, more generally, given any two terms `a_1r^m` and `a_1r^n`, `m < n`, we can find `r` by dividing `(a_1r^n)/(a_1r^m)` and taking the `(n-m)^"th"` root:

`((a_1r^n)/(a_1r^m))^(1/(n-m)) = (r^(n-m))^(1/(n-m)) = r^((n-m)/(n-m))=r^1=r`