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

MATHEMATICS · Multi-part · A Level · 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}.