General (nth) Term of a Geometric Progression

Deriving the General (nth) Term

Let the first term of a GP be $a$ and the common ratio be $r$. Then the terms are:

Term 1 $= a = ar^{0}$, Term 2 $= ar = ar^{1}$, Term 3 $= ar^2$, and so on. In general, the nth term of a GP is:

This single formula lets us find any term of a GP directly, without writing out every term before it.

Worked idea: to find the 5th term of the GP $3, 6, 12, \ldots$: here $a = 3$, $r = 2$, $n = 5$, so $T_5 = 3 \times 2^{4} = 3 \times 16 = 48$.

Finding an Unknown First Term or Common Ratio …