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Geometric progressions Cambridge IGCSE Additional Mathematics revision

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In plain words

3, 6, 12, 24: multiply by 2 each time. A sequence where each term is the last one times a fixed number is a geometric progression. If that number is a fraction, the terms shrink, and something remarkable happens: the sum of all of them, for ever, is a finite number.

6 things to know

  1. In a geometric progression, each term is the one before multiplied by the common ratio, r. The first term is a.
  2. The nth term is arⁿ⁻¹.
  3. The sum of the first n terms is Sₙ = a(1 − rⁿ) ÷ (1 − r), for r ≠ 1.
  4. If r is between −1 and 1, the progression converges, and it has a sum to infinity: S∞ = a ÷ (1 − r).
  5. If r is 1 or more, or −1 or less, the terms do not shrink towards zero, and there is no sum to infinity.
  6. To find r, divide any term by the one before it.

Worked example

For the progression 3, 6, 12, …, find the 8th term and the sum of the first 8 terms.

  1. a = 3 and r = 2.
  2. 8th term = 3 × 2⁷ = 384.
  3. S₈ = 3(1 − 2⁸) ÷ (1 − 2) = 3 × 255 = 765.

Worked example

Find the sum to infinity of 16 + 8 + 4 + ….

  1. a = 16 and r = 1/2. Since r is between −1 and 1, the sum to infinity exists.
  2. S∞ = 16 ÷ (1 − 1/2) = 32.

Tips and tricks

  • The nth term has the power n − 1, not n. The 8th term uses r⁷.
  • Before using the sum to infinity, state that |r| < 1. A question may ask you to explain why a sum to infinity does or does not exist.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Geometric progressions: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is the common ratio of 4, 12, 36, 108, …?
    • 8
    • 3 (the answer)
    • 1/3
    • 4

    Each term is 3 times the one before.

  2. What is the 5th term of the progression with a = 2 and r = 3?
    • 54
    • 162 (the answer)
    • 486
    • 30

    2 × 3⁴.

  3. When does a geometric progression have a sum to infinity?
    • when r is greater than 1
    • when r is between −1 and 1 (the answer)
    • when a is positive
    • always

    The terms must shrink towards zero.

  4. What is the sum to infinity of 10 + 5 + 2.5 + …?
    • 15
    • 17.5
    • 20 (the answer)
    • infinite

    10 ÷ (1 − 1/2).

  5. Which sequence is a geometric progression?
    • 2, 4, 6, 8
    • 5, 10, 20, 40 (the answer)
    • 1, 4, 9, 16
    • 3, 6, 10, 15

    It has a constant ratio of 2.

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