medium difficulty5 min

Geometric Progression — Definition

A geometric progression is a sequence in which each term is obtained by multiplying the previous term by the same non-zero constant ratio.

In one line: GP means same multiplier every step.

Practice This Concept

Understand it

In a GP, the common ratio r stays the same. If the first term is a, then the nth term is an = a · r^(n - 1). GP patterns appear in repeated doubling, halving, compound growth, and certain decay situations.

Example

3, 6, 12, 24 is a GP with first term a = 3 and common ratio r = 2.

Formula and rule

For a GP, r = next term ÷ previous term and an = a · r^(n - 1).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Find the 5th term of 2, 6, 18, ... .
2Here a = 2, r = 3. a5 = 2 × 3⁴ = 2 × 81 = 162.

Method you can reuse

1Check ratios of consecutive terms.
2Confirm the ratio is constant.
3Identify a and r.
4Use an = a · r^(n - 1).
5Handle fractional or negative ratios carefully.

Visual tip

A repeated multiplication arrow diagram can show each term being multiplied by r.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is 81, 27, 9, 3 a GP? State the common ratio.

Think first, then show answer

Yes, it is a GP because each term is obtained by multiplying the previous term by 1/3, so the common ratio is r = 1/3.

Common wrong answer

For 2, 6, 18, students may use d = 4 and treat it as AP, but the difference is not constant while the ratio is 3.

Don't mix this up

Students sometimes subtract consecutive terms and call it GP, but GP depends on ratio, not difference.

Exam tip

To test a GP, divide consecutive terms: second by first, third by second, and so on.