## Geometric Sequences and Series

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

### Geometric Sequences

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 = a r 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 an geometric sequence of positive terms is 8 and the fifth term 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 = a r n-1 32 = ar 5 − 1 8 = ar 3 − 1 32 = ar 48 = ar 2 4 = r2              ( dividing)r = ±2                 The common ratio is 2 (discard -2) 8 = a x 22a = 2                 the first term The sequence is 2, 4, 8, 16, ...

### Geometric Series

If terms of a geometric sequence are added together a geometric series is formed.

2 + 4 + 8 + 16 is a finite geometric series
2 + 4 + 8 + 16 + ... is an infinte geometric series

To find the sum of the first n terms of a geometric sequence use the formula:

 Sum of first n terms of a geometric sequence wherer = common ratioa = first termn = number of terms OR

If the common ratio is a fraction i.e. -1 < r < 1 then an equivalent formula, shown below is easier to use.

 Sum of first n terms of a geometric sequence for when -1 < r < 1 i.e. r is a fraction Example What is the sum of the first 10 terms of the geometric sequence:       3, 6, 12, ...

 Common ratio r = 6 ÷ 3 = 2 Number of terms n = 10 First term a = 3 ### The Sum to Infinity of a Geometric Sequence

Spreadsheets are very useful for generating sequences and series.

For a geometric sequence with a common ratio greater than 1: The formula in cell B3 is = B2*2 The formula in cell D3 is =D2 + B3 The fill down command is then used to complete the sequences.

It can be seen that as successive terms are added the sum of the terms increases.
If there were an infinite number of terms the sum would be infinity.

For a geometric sequence with a common ratio less than 1: The formula in cell B3 is = B2*2 The formula in cell D3 is =D2 + B3 The fill down command is then used to comlete the sequences.

It can be seen that as successive terms are added the sum of the terms appears to be heading towards 16.
If there were an infinite number of terms the sum would be 16.

This is called the sum to infinity of a geometric sequence and only applies when the common ratio is a fraction

i.e. -1 < r < +1.
r
can be positive or negative.

The following formula can be used:

 Sum to infinity of geometric sequence wherer = common ratioa = first term Example

Find the sum to infinity of the geometric sequence 8, 4, 2, 1, ...

a = 8 and r = 0.5 As can be seen from cell D10 in the spreadsheet above, 16 is the value the sums were heading towards.

To see this concept clearly illustrated - 