Geometric sequences and series

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What this lesson covers

  • The nth term is arⁿ⁻¹.
  • The sum of n terms is a(rⁿ − 1) ÷ (r − 1).
  • If r is between −1 and 1, the infinite sum is a ÷ (1 − r).
Key idea

The three formulas

  • a is the first term and r is the common ratio.
  • tₙ = arⁿ⁻¹.
  • Sₙ = a(rⁿ − 1) ÷ (r − 1).
  • S∞ = a ÷ (1 − r), only when −1 < r < 1.
Worked example

3, 6, 12, ...

  • Find t₈ and S₈.
  1. a = 3, r = 2
  2. t₈ = 3 × 2⁷ = 384
  3. S₈ = 3(2⁸ − 1) ÷ 1
  4. S₈ = 765
Worked example

40, 20, 10, ...

  • The ratio is a half, so an infinite sum exists.
  1. r = 1/2
  2. S∞ = 40 ÷ (1 − 1/2)
  3. S∞ = 80

Common mistakes

  • Writing 3 × 2⁸ for the eighth term.
    The power is one less than the term number.
  • Using S∞ when r = 2.
    The terms grow, so the sum has no limit.

Your turn

  • 1. For 2, 6, 18, ... find t₆.
  • 2. Find S₆.
  • 3. Find S∞ for 81, 27, 9, ...
Show answer

1. 2 × 3⁵ = 486 2. 2(729 − 1) ÷ 2 = 728 3. 81 ÷ (2/3) = 121.5

Summary

What to take away

  • Same ratio means geometric.
  • Check −1 < r < 1 before using S∞.

More in AS 91258

Arithmetic sequences and seriesNotesGeometric sequences and seriesNotesSigma notationNotesGrowth and financial problemsNotes

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