# Sum of Geometric Sequence/Examples/One Seventh from 1 to n

## Theorem

$\ds \sum_{j \mathop = 0}^n \dfrac 1 {7^j} = \frac 7 6 \paren {1 - \frac 1 {7^{n + 1} } }$

## Proof

 $\ds \sum_{j \mathop = 0}^n \dfrac 1 {7^j}$ $=$ $\ds \frac {1 - \frac 1 7^{n + 1} } {1 - \frac 1 7}$ $\ds$ $=$ $\ds \frac {7 - 7 \frac 1 7^{n + 1} } {7 - 1}$ multiplying top and bottom by $7$

Hence the result.

$\blacksquare$