Write the recurring decimal 0.363636... as a common fraction.
- 0.363636... = 0.36 + 0.0036 + 0.000036 + ..., a G.P. with a = 0.36 and r = 0.01.
- |r| < 1, so S∞ = a/(1 − r) = 0.36/(1 − 0.01) = 0.36/0.99.
- = 36/99 = 4/11.
Answer: 0.363636... = 4/11
Geometric progressions
Maths · Sequences and series. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A geometric progression (G.P.) is a sequence in which each term is found by multiplying the term before it by the same non-zero number. That number is the common ratio r, and the first term is a. No term of a G.P. is zero.
A geometric mean (G.M.) is a number placed between two numbers so that all of them are in G.P. If a, b, c are in G.P., then b2 = ac.
When |r| < 1 the terms get smaller and smaller, so the sum of infinitely many terms settles at a fixed value, the sum to infinity. A recurring decimal is a series of this kind, which is how it is changed into a common fraction.
Rules to remember
- common ratio = any term ÷ the term before it (r = an/an−1)
- nth term: an = arn−1
- G.M. between a and b: G2 = ab, so G = ±√(ab)
- n geometric means between a and b: rn+1 = b/a, then the means are ar, ar2, ...
- sum of n terms: Sn = a(1 − rn)/(1 − r), which is the same as a(rn − 1)/(r − 1), for r ≠ 1
- sum to infinity: S∞ = a/(1 − r), only when |r| < 1
Common mistake
Students use a/(1 − r) when |r| ≥ 1. An infinite geometric series has a sum only when |r| < 1, so check r first. Also keep the sign of a negative r inside brackets when you raise it to a power.
Worked example
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