Geometric sequences
Objective. Use common ratios and powers to describe geometric growth.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
A geometric sequence multiplies each term by a fixed common ratio r to get the next term, rather than adding a fixed amount. The nth term is un = a·r^(n−1), where a is the first term. To find r, divide any term by the term before it: for 3, 6, 12, 24, r = 6÷3 = 2, so un = 3·2^(n−1). To find a later term, raise r to the appropriate power rather than repeatedly multiplying by hand when n is large. Always confirm r is consistent by checking it between two different pairs of consecutive terms.
Worked example
Find the 6th term of the geometric sequence 5, 10, 20, 40, …
- 1Common ratio r = 10÷5 = 2, first term a = 5
- 2Formula: un = 5 × 2^(n−1)
- 3For n = 6: u6 = 5 × 2^5
- 42^5 = 32, so u6 = 5 × 32 = 160
Answer: 160
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Start practisingCommon mistake. Pupils treat a geometric sequence as arithmetic and add a constant difference instead of multiplying by r. For 2, 6, 18, 54 they wrongly predict the next term as 54+16=70 instead of 54×3=162. Always test by dividing consecutive terms for a constant ratio first.
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