Geometric progression

Geometric progression shortly known as GP (also in accurately known as geometric series)is a sequence of numbers such that the quotient of any two successive numbers of the sequence is always a constant. That constant is called as the common ratio of the sequence.

a, ar, ar2, ar3, ar4,....arn-1,... is a GP.

Here a or a1 is the first term and ris the common ratio. ar is the second term known as a2, ar2 is the third term known as a3, .... and arn-1 is known as nth term.

Example:

3,9,27,81,243,.....

Here a=3,

r= a2/a = a3/a2 = 9/3 = 27/9 =3


Formula to find the nth term of a GP

an=arn-1

Example:

Find the 25th term of the progression, 2,4,8,16,32,...

Solution:

a=2, r = 4/2 = 2.

       a25 = 2x225-1

       a25 = 2x224

       a25 = 2x16777216

            = 33554432


The sum of Geometric progression:

Sum to n terms of a GP means: a+ar+ar2+...arn-1 and it is denoted by Sn

Sum of infinite geometric serieswhen r<1 is

Sn = 1/1-r which is valid only when |r|<1

Example:

Find S10 200, 100, 50, 25,...

Solution:

Here a = 200 and r= 100/200 =1/2 = 0.5.

So S10 = 200(1-(0.5)10)/1-0.5

             = 200x0.9990234375/0.5

             = 199.80468/0.5

             = 399.609375






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