What is the common ratio of the sequence whose first term and third terms are 2 and 32 respectively

Given a geometric sequence with the first term a 1 and the common ratio r , the n th (or general) term is given by
a n = a 1 ⋅ r n − 1 .

Example 1:

Find the 6 th term in the geometric sequence 3 , 12 , 48 , ... .

a 1 = 3 ,     r = 12 3 = 4 a 6 = 3 ⋅ 4 6 − 1 = 3 ⋅ 4 5 = 3072

Example 2:

Find the 7 th term for the geometric sequence in which a 2 = 24 and a 5 = 3 .

Substitute 24 for a 2 and 3 for a 5 in the formula

a n = a 1 ⋅ r n − 1 .

a 2 = a 1 ⋅ r 2 − 1 → 24 = a 1 r a 5 = a 1 ⋅ r 5 − 1 →         3 = a 1 r 4

Solve the firstequation for a 1 : a 1 = 24 r

Substitute this expression for a 1 in the second equation and solve for r .

3 = 24 r ⋅ r 4 3 = 24 r 3 1 8 = r 3   so   r = 1 2

Substitute for r in the first equation and solve for a 1 .

24 = a 1 ( 1 2 ) 48 = a 1

Now use the formula to find a 7 .

a 7 = 48 ( 1 2 ) 7 − 1 = 48 ⋅ 1 64 = 3 4

See also: sigma notation of a series and n th term of a arithmetic sequence

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 ratio, r.
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.

General Term, tn

A geometric sequence can be written:

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 term n = 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 ratio a = 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 -