General Term of a G.P.

The General Term of a Geometric Progression

A geometric progression (G.P.) is defined by two things: its first term and the fixed multiplier that takes you from one term to the next. The first term is denoted by $a$, and it must be non-zero (otherwise every term would be zero, which is a trivial case). The fixed multiplier is called the common ratio, denoted by $r$.

To see the pattern, write out the first few terms explicitly.

The first term is simply $a$.

The second term is the first term multiplied by $r$: $a_2 = a \cdot r = ar$.

The third term is the second term multiplied by $r$: $a_3 = a_2 \cdot r = (ar) \cdot r = ar^2$.

The fourth term is the third term multiplied by $r$: $a_4 = a_3 \cdot r = (ar^2) \cdot r = ar^3$.

Continuing this way, the fifth term is $a_5 = ar^4$, the sixth term is $a_6 = ar^5$, and so on.

Now look at the pattern in the exponents of $r$:

• $a_1 = a = a \cdot r^{0} = a \cdot r^{1-1}$
• $a_2 = ar = a \cdot r^{1} = a \cdot r^{2-1}$
• $a_3 = ar^2 = a \cdot r^{2} = a \cdot r^{3-1}$
• $a_4 = ar^3 = a \cdot r^{3} = a \cdot r^{4-1}$
• $a_5 = ar^4 = a \cdot r^{4} = a \cdot r^{5-1}$

In each case, the exponent of $r$ is exactly one less than the position number of the term. So for the $n$th term, the exponent of $r$ is $n-1$.

This is the general term (or $n$th term) of a geometric progression with first term $a$ and common ratio $r$.

Finite and Infinite Geometric Progressions

Using the general term, we can write the entire G.P. in a compact form.

• A finite G.P. with $n$ terms is written as:

• An infinite G.P. (which continues without end) is written as: $a, \; ar, \; ar^2, \; ar^3, \; \dots, \; ar^{n-1}, \; \dots$ …