Question

In a geometric sequence `t_3 = 45` `t_6=1215`, how do you find the first term?

Answer 1

`t_1=5`

Explanation:

For any geometric sequence, the common ratio between successive terms has value `r=(t_n+1)/t_n`.

From this we can conclude that `t_n=r^(n-1)*t_1`

Hence if we work from term 3 to term 6 we may write that

`r^3t_3=t_6`

`therefore 45r^3=1215`

`therefore r=root(3)(1215/45)=3`.

This implies that `t_1=t_3/r^2=45/9=5`