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Algebra · Higher

Geometric sequences

In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio (r). These are distinct from arithmetic sequences where a fixed amount is added.

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Key facts to remember

  • 1Common ratio r = (any term) ÷ (previous term).
  • 2nth term formula: aₙ = a × rⁿ⁻¹ where a is the first term.
  • 3If |r| > 1 the sequence is growing (divergent); if |r| < 1 it is shrinking (convergent).
  • 4If r is negative, terms alternate in sign.
  • 5Geometric sequences appear in compound interest and exponential growth problems.
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Formulas

nth term
aₙ = a × rⁿ⁻¹

a = first term, r = common ratio

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Worked examples

Example 1

Find the 6th term of the geometric sequence 3, 6, 12, 24, …

Working

  1. Common ratio r = 6 ÷ 3 = 2
  2. nth term formula: aₙ = 3 × 2ⁿ⁻¹
  3. a₆ = 3 × 2⁵ = 3 × 32 = 96
Answer96
Example 2

The 2nd term of a geometric sequence is 12 and the 4th term is 108. Find the common ratio and the 1st term.

Working

  1. a × r = 12 and a × r³ = 108
  2. Divide: r² = 108 ÷ 12 = 9, so r = 3
  3. a = 12 ÷ 3 = 4
  4. 1st term = 4
Answerr = 3, first term = 4
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Common mistakes

✗Confusing geometric (multiply) with arithmetic (add) sequences.
✗Using the wrong formula: using the arithmetic nth term a + (n−1)d for a geometric sequence.
✗Not checking whether r is negative when alternating signs are present.
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Exam tips

✓Calculate r by dividing consecutive terms — check it is constant for two or more pairs.
✓Be careful with the index: the nth term uses rⁿ⁻¹, not rⁿ.

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