Give a recursive definition of the geometric sequence with i

Give a recursive definition of the geometric sequence with initial term a and common ratio r. Make sure you write down a recursive formula: (1) give the initial term, and (2) give the rule for building new terms from previous terms.

Solution

Consider the geometric sequence

                                                      a, ar, ar²,........ar^n-1,....

Recursive definition of the nth term:

                                                        a_n = ra_{n-1 ,} n=2,3... with a₁ =a

(a₁ =a gives the initial term and the recursive definition

                                                        a_n = ra_n-1, n=2,3...

builds the remaining terms:

for example

                                                   a₃ = ra₂

                                                                   = r r a₁

                                                       =ar² )