Originally Posted by

**Grandad** Hello brentwoodbcTo find the number of terms ($\displaystyle n$) of a GP, if you know the first and last terms, use the fact that the $\displaystyle n^{th}$ term is $\displaystyle ar^{n-1}$, where $\displaystyle a$ is the first term and $\displaystyle r$ is the common ratio.

For example, we can find the number of terms in the GP $\displaystyle 3 + 15 + 75 + ... + 732421875$, by saying $\displaystyle a = 3, r = 5$ and $\displaystyle ar^{n-1} = 732421875$

$\displaystyle \Rightarrow 3 \times 5^{n-1} = 732421875$

$\displaystyle \Rightarrow 5^{n-1} = \frac{732421875}{3} = 244140625$

Now take logs (to any base) of both sides:

$\displaystyle (n-1)\log 5 = \log 244140625$

$\displaystyle \Rightarrow n = 1 + \frac{\log 244140625}{\log 5} = 13$

Does that answer your question?

Grandad