Geometric progressions: the nth term

6 minAS Level1.6.21.6.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can recognise a geometric progression, find any term with $u_n = ar^{n-1}$ , and use $b^2 = ac$ to test whether three numbers are in geometric progression.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$u_n$	the $n$ th term (so $u_1$ is the first, $u_2$ the second, …)
$u_{n+1}$	the term straight after $u_n$
$a$	the first term, $u_1$
$r$	the common ratio: the number you multiply by each time

Multiply, don't add

Fold a sheet of paper in half: 2 layers. Again: 4. Again: 8. Each fold doubles the layers.

A geometric progression (GP) works like that: you get each term by multiplying the previous one by the same number.

That number is the common ratio $r$ . The first term is $a$ , as before. Example: $2, 4, 8, 16, \dots$ has $a = 2$ and $r = 2$ .

APs grow at a steady pace. GPs explode (or shrink away to nothing). That's the whole drama of this topic.

Definition

Geometric progression

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a constant, the common ratio $r$ (with $a \neq 0$ and $r \neq 0$ ):

$\frac{u_{n+1}}{u_n} = r \quad \text{for every } n.$

The $n$ th term is $u_n = ar^{n-1}$ .

Why the power is n − 1

Write out the first few terms:

$u_1 = a$ (no multiplying yet)
$u_2 = ar$ (multiplied by $r$ once)
$u_3 = ar \times r = ar^2$ (twice)
$u_4 = ar^3$ (three times)

To reach the $n$ th term you multiply by $r$ exactly $n - 1$ times. It's the same "steps, not terms" idea as in an AP.

To recognise a GP, divide each term by the one before. If the ratio is the same every time, it's geometric.

Quick check

Fill in the blanksQuestion 1 · Recall

Multiple choiceQuestion 2 · Recall

Answer the 3 questions above to continue.