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.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $12$