Consider a geometric sequence with first term 1 and common ratio 10

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Consider a geometric sequence with first term 1 and common ratio 10.

S_n

is the sum of the first

n

terms of the sequence. (1) Find an expression for

S_n

in the form

\frac{a^n-1}{b}

, where

a,b\in\mathbb{Z}^+

. (2) Hence, show that

S_1+S_2+S_3+\dots+S_n=\frac{10(10^n-1)-9n}{81}.