A geometric sequence has u1 = 40 and u4 = 5. Find the sum to infinity.
- u4 ÷ u1 = r3, so r3 = 5 ÷ 40 = 1/8 and r = 1/2.
- |r| < 1, so the sum to infinity exists.
- S∞ = u1/(1 − r) = 40/(1 − 1/2) = 40 ÷ 0.5.
Answer: S∞ = 80
Geometric sequences, series and financial applications
Maths: Analysis and Approaches · Number and algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
In a geometric sequence each term is the one before it multiplied by the same number, the common ratio r. To reach the nth term you multiply the first term by r a total of (n − 1) times.
If r lies between −1 and 1, the terms shrink towards zero and the sum settles at a fixed value, the sum to infinity. If |r| ≥ 1 there is no sum to infinity.
Money problems are geometric. Compound interest of 5% a year means multiplying by 1.05 each year. Depreciation of 15% a year means multiplying by 0.85 each year. Growth and decay in science work the same way.
Rules to remember
- nth term = first term × ratio to the power (n − 1) (un = u1 × rn − 1)
- Common ratio: r = u2 ÷ u1 = u3 ÷ u2
- Sum of the first n terms: Sn = u1(rn − 1)/(r − 1) = u1(1 − rn)/(1 − r), r ≠ 1
- Sum to infinity: S∞ = u1/(1 − r), only when |r| < 1
- Two terms given: um ÷ uk = rm − k
- Compound interest: FV = PV × (1 + r/(100k))kn, with r% the nominal annual rate, k compounding periods per year, n years
- Depreciation at r% a year for n years: value = start value × (1 − r/100)n
Common mistake
Using the sum to infinity formula without checking that |r| < 1, or writing the power as n instead of n − 1 in the nth term.
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