Geometric sequence


A geometric sequence (or geometric progression) is a number pattern in which there's a common ratio between successive terms.


Example of a geometric sequence


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

rˉ¹ = 3² = 3 x 3 = 9

3rd term  =  1  x  9  =  9


1st term = 1

r = 3,   n - 1 = 5 - 1 = 4

rˉ¹ = 3= 3 x 3 x 3 x 3 = 81

5th term  =  1  x  81  =  81


 

The common ratio can also be less than one


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 rˉ¹

1st term = 81

r = 1/3,   n - 1 = 4 - 1 = 3

rˉ¹ = (1/3)³ = (1 x 1 x 1)/(3 x 3 x 3) = 1/27

4th term  =  81  x  1/27  =  3


 

Some more examples

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 rˉ¹

1st term = 1

r = 2,   n - 1 = 9 - 1 = 8

rˉ¹ = 2 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

9th term  =  1  x  256  =  256




Try working out the 8th term in this number pattern

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 rˉ¹

1st term = 256

r = 1/2,   n - 1 = 8 - 1 = 7

rⁿˉ¹ = (1/2) = (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



Summary of geometric sequences


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 rˉ¹


Not learned about arithmetic sequences yet? Click here.


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