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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

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.

Example

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

Rule to remember

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

Memory hook

GP means same multiplier every step.

Examples and method

Worked example

Find the 5th term of 2, 6, 18, ... . Here a = 2, r = 3. a5 = 2 × 3^4 = 2 × 81 = 162.

Method to apply

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

Diagram support

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

How CBSE asks it

Students may be asked to identify GP, find common ratio, find nth term, or compare AP and GP behaviour.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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.

Study the geometric progression — definition diagram carefully

Use the labelled diagram to keep geometric progression — definition clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of geometric progression — definition in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on geometric progression — definition.

Revision cue

Revise geometric progression — definition through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

A geometric progression is a sequence in which each term is obtained by multiplying the previous term by the same non-zero constant ratio. For a GP, r = next term ÷ previous term and an = a · r^(n - 1). 3, 6, 12, 24 is a GP with first term a = 3 and common ratio r = 2.

3-mark answer

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. For a GP, r = next term ÷ previous term and an = a · r^(n - 1). Find the 5th term of 2, 6, 18, ... . Here a = 2, r = 3. a5 = 2 × 3^4 = 2 × 81 = 162. Students may be asked to identify GP, find common ratio, find nth term, or compare AP and GP behaviour. 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.
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