## Geometric Sequences

geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number.

### Notation

This fixed number is called the common ratior.
The common ratio can be positive or negative, an integer or a fraction.

The common ratio can be calculated by dividing any term by the one before it.
e.g. Common ratio = t n+1 ÷ t n

The first term of a geometric sequence is shown by the variable a.

 Geometric Sequence First term, a Common ratio, r 2, 6, 18, 54, ... 2 6 ÷ 2 = 318 ÷ 6 = 3        etc. 20, 10, 5, 2.5, ... 20 10 ÷ 20 = 0.5 5 ÷ 10 = 0.5      etc.

### General Term, tn

A geometric sequence can be written:

 First term Second term Third term Fourth term General term (n th term) t 1 t 2 t 3 t 4 ... t n a ar ar2 ar3 ... ar n − 1

 Example 1 What is the common ratio of the geometric sequence: 20, 10, 5, 2.5,, ... Common ratio = t n+1 ÷ t n Second term ÷ first term = t2 ÷ t1 = 10 ÷ 20 = 0.5 Check: Third term ÷ second term = 5 ÷ 10 = 0.5 The common ratio is 0.5 Example 2 Find the 8th term of the geometric sequence: 3, 9, 27, 81, ... Common ratio, r = 9 ÷ 3 = 3First term a = 3 using tn = ar n − 1 t8 = 3 x 3 8- 1= 3 x 3 7= 6561 The 8th term is 6561 Example 3 Which term of the sequence 2, 4, 8, 16, ... would be equal to 1024? Common ratio, r = 2First term, a = 2 Using tn = ar n-1 1024 =2 x 2 n-11024 = 21 x 2 n − 11024 = 2 nn = 10 1024 is the 10th term. Example 4 The third term of a geometric sequence is 8 and the fifthterm is 32. Find the first term, a, and the common ratio, r, and thus list the first four terms of the sequence. t 3 = 8t 5 = 32 using tn = ar n-1 32 = ar 5 − 1 8 = ar 3 − 1 32 = ar 48 = ar 2          ( dividing) 4 = r2                      r = 2 or -2                the common ratio 8 = a x 22a = 2                 the first term The sequence is 2, 4, 8, 16, ... or 2, -4, 8, -16...