Math · Precalculus · Worked example
Decide if a sequence is arithmetic or geometric
Decide whether each sequence is arithmetic, geometric or neither, and give a formula for its nth term: (a) 2, 6, 18, 54, …; (b) 7, 3, −1, −5, …; (c) 1, 4, 9, 16, ….
Test (a)
The differences 4, 12 and 36 change, but the ratios 6/2, 18/6 and 54/18 are all 3. So (a) is geometric with a₁ = 2 and r = 3.
Test (b)
The differences 3 − 7, −1 − 3 and −5 − (−1) are all −4. So (b) is arithmetic with a₁ = 7 and d = −4.
Test (c)
The differences 3, 5 and 7 change, and so do the ratios 4, 9/4 and 16/9, so (c) is neither. Its terms are the squares: aₙ = n².
Write them recursively
The same sequences can be built one step at a time: (a) a₁ = 2 and aₙ₊₁ = 3aₙ; (b) a₁ = 7 and aₙ₊₁ = aₙ − 4.
Result
(a) Geometric, aₙ = 2 · 3ⁿ⁻¹. (b) Arithmetic, aₙ = 11 − 4n. (c) Neither: aₙ = n².
Your turn
Is 100, 90, 81, 72.9, … arithmetic or geometric? Give its nth term.
Show the answer and explanation
Geometric with r = 0.9, so aₙ = 100(0.9)ⁿ⁻¹.
The differences −10, −9 and −8.1 change, but each ratio is 0.9.
Keep exploring
In Sequences & infinite series, the recurrence with next term 3a and first term 2 lists 2, 6, 18, 54, …. Change 3a to a − 4 and the first term to 7 to build sequence (b).
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