Question

The formula for the sum of the first n terms of a geometric sequences `S_n=5^(n-1)` how do you find the first four terms of the sequence?

Answer 1

The terms are, `t_1=4/5, t_2=4, t_3=20, and, t_4=100`.

Explanation:

Given that, `S_n=5^(n-1)=5^n/5`.

Observe that, in any sequence `{t_n : n in NN},`

`S_n=ul(t_1+t_2+t_3+...+t_(n-1))+t_n`

`=S_(n-1)+t_n`

`rArr t_n=S_n-S_(n-1)`.

Using this Formula in our case, we get,

`t_n=5^n/5-5^(n-1)/5=5^n/5-5^n/25`

`=5^n/25(5-1)=4*5^(n-2)`

Accordingly, we have,

`t_1=4/5, t_2=4, t_3=20, and, t_4=100`.