# 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?

Aug 22, 2016

The terms are, ${t}_{1} = \frac{4}{5} , {t}_{2} = 4 , {t}_{3} = 20 , \mathmr{and} , {t}_{4} = 100$.

#### Explanation:

Given that, ${S}_{n} = {5}^{n - 1} = {5}^{n} / 5$.

Observe that, in any sequence $\left\{{t}_{n} : n \in \mathbb{N}\right\} ,$

${S}_{n} = \underline{{t}_{1} + {t}_{2} + {t}_{3} + \ldots + {t}_{n - 1}} + {t}_{n}$

$= {S}_{n - 1} + {t}_{n}$

$\Rightarrow {t}_{n} = {S}_{n} - {S}_{n - 1}$.

Using this Formula in our case, we get,

${t}_{n} = {5}^{n} / 5 - {5}^{n - 1} / 5 = {5}^{n} / 5 - {5}^{n} / 25$

$= {5}^{n} / 25 \left(5 - 1\right) = 4 \cdot {5}^{n - 2}$

Accordingly, we have,

${t}_{1} = \frac{4}{5} , {t}_{2} = 4 , {t}_{3} = 20 , \mathmr{and} , {t}_{4} = 100$.