learning geometric progression

Question

Learning geometric progression vis-á-vis arithmetic progression should make it easier. In geometric progression also we denote the first term by 'a' but a  basic difference from A.P. is that instead of common difference we have common ratio 'r'. Like d, r remains constant whenever the ratio of any two consecutive terms is computed. The terms of a G.P. are

                   a, ar, ar², ar³, ar⁴, ................, ar^{n - 1}  

That is,         T₁     =     a

                   T₂     =     ar

                   T₃     =     ar²

                   :                :
                   :                :

                   T_n     =     ar^{n - 1}

This is similar to A.P. We take an example to become more familiar with this.

Example 

It is known that the first term in G.P. is 3 and the common ratio r is 2. Find the first three terms of this series and also the nth term.

We know that the first term is given by

                   T₁     = a   = 3

                   T₂     = ar   = 3.2     = 6

                   T₃     = ar²  = 3.2.2 = 12

The nth term is given by  T_n  = ar^n-1  = 3(2)^n-1