Arithmetic progressions
An arithmetic progression (AP) is a sequence in which each term is found by adding a fixed number, the common difference d, to the previous term. If the first term is a, the terms are a, a+d, a+2d, ...
Key idea
nth term: Tn = a + (n − 1)d. Sum of n terms: Sn = n⁄2[2a + (n − 1)d] = n⁄2(a + l), where l is the last term.
Geometric progressions
A geometric progression (GP) multiplies each term by a fixed common ratio r. The terms are a, ar, ar2, ... The nth term is Tn = arn−1 and the sum of n terms is Sn = a(rn − 1)⁄(r − 1) for r ≠ 1.
When |r| < 1 the terms shrink towards zero and the series has a finite sum to infinity, S∞ = a⁄(1 − r). If |r| ≥ 1 no sum to infinity exists.
Worked example
For the GP 24, 12, 6, ... we have a = 24 and r = ½. The sum to infinity is S∞ = 24⁄(1 − ½) = 24⁄(½) = 48.
Remember
- Check whether a sequence is arithmetic (constant difference) or geometric (constant ratio) first.
- The sum to infinity only exists when |r| < 1.
- Use Tn = a + (n − 1)d, not a + nd.