A geometric sequence (or geometric progression) is a number pattern in which there's a common ratio between successive terms.
Take a look at the number pattern
1 |
3 |
9 |
27 |
81 |
Closer inspection reveals a common ratio of 3
1 |
x3 |
3 |
x3 |
9 |
x3 |
27 |
x3 |
81 |
To find the nth term of a geometric progression you can use the formula
where r = common ratio.
Have a go at testing this formula on the 3rd and 5th terms of the above sequence
1st term = 1
r = 3, n - 1 = 3 - 1 = 2
rⁿˉ¹ = 3² = 3 x 3 = 9
3rd term = 1 x 9 = 9
1st term = 1
r = 3, n - 1 = 5 - 1 = 4
rⁿˉ¹ = 3⁴ = 3 x 3 x 3 x 3 = 81
5th term = 1 x 81 = 81
For instance, if you reverse the above progression you get
81 |
27 |
9 |
3 |
1 |
common ratio 1/3 (a third)
81 |
x1/3 |
27 |
x1/3 |
9 |
x1/3 |
3 |
x1/3 |
1 |
Try applying the nth term formula to the 4th term
nth term = 1st term x rⁿˉ¹
1st term = 81
r = 1/3, n - 1 = 4 - 1 = 3
rⁿˉ¹ = (1/3)³ = (1 x 1 x 1)/(3 x 3 x 3) = 1/27
4th term = 81 x 1/27 = 3
1 |
2 |
4 |
8 |
16 |
In this progression, the common ratio is 2
1 |
x2 |
2 |
x2 |
4 |
x2 |
8 |
x2 |
16 |
To find say the 9th term, you could continue writing down the sequence until you reached this term
1, 2, 4, 8, 16, 32, 64, 128, 256
But you can also use the formula for the nth term
nth term = 1st term x rⁿˉ¹
1st term = 1
r = 2, n - 1 = 9 - 1 = 8
rⁿˉ¹ = 2⁸ = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256
9th term = 1 x 256 = 256
256 |
128 |
64 |
32 |
16 |
The common ratio's 1/2
256 |
x1/2 |
128 |
x1/2 |
64 |
x1/2 |
32 |
x1/2 |
16 |
Your calculation should look similar to this
nth term = 1st term x rⁿˉ¹
1st term = 256
r = 1/2, n - 1 = 8 - 1 = 7
rⁿˉ¹ = (1/2)⁷ = (1 x 1 x 1 x 1 x 1 x 1 x 1 x 1)/(2 x 2 x 2 x 2 x 2 x 2 x 2) = 1/128
8th term = 256 x 1/128 = 2
The nth term of a geometric progression can be found by plugging the 1st term and common ratio (r) values into the geometric sequence formula nth term = 1st term x rⁿˉ¹
Not learned about arithmetic sequences yet? Click here.