A geometric sequence has u1 = 5 and r = 3. Find the sum of the first 7 terms.
- Use Sn = u1(rn − 1)/(r − 1) with n = 7.
- 37 = 2187, so r7 − 1 = 2186.
- S7 = 5 × 2186 ÷ (3 − 1) = 10 930 ÷ 2 = 5465.
Answer: S7 = 5465
Geometric sequences and series
Maths: Applications and Interpretation · 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 you multiply by the same number each time to get the next term. That number is the common ratio, r. If r is bigger than 1 the terms grow. If r is between 0 and 1 the terms shrink.
Percentage change gives a geometric sequence. A rise of 4% each year means r = 1.04, and a fall of 30% means r = 0.7. Population growth, salaries with a yearly percentage rise, bouncing balls and doubling bacteria are all geometric.
Read the question carefully to decide whether it asks for one term (the value in a given year) or a sum (the total over several years).
Rules to remember
- common ratio = any term ÷ the term before it (r = un+1 ÷ un)
- nth term: un = u1 × rn − 1
- Sum of the first n terms: Sn = u1(rn − 1)/(r − 1), for r ≠ 1
- Same sum, easier when r < 1: Sn = u1(1 − rn)/(1 − r)
- Increase of p% per period: r = 1 + p/100. Decrease of p%: r = 1 − p/100
- Given two terms, divide them to find r: u5 ÷ u2 = r3
- Value after n periods of growth = starting value × rn, because the starting value is at time 0
Common mistake
Students mix up the power. The nth term uses rn − 1, but a quantity after n years of growth uses rn. Count how many times the multiplier has been applied.
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