Answer to Question #315069 in Calculus for Shanu

Question

Represent the sum of first terms of the series 3+33+333+.........

using the sigma notation.

Answer 1

$$S= 3(1+11+111+1111+...n^{th} term).$$

Let the n^th term be,

$tn=1+10+100+...10^{n-1}$

This is a G.P. with first term 1 and common ratio of 10.

Thus, the sum of n terms,

$S= 3\sum_{i=1}^{n} t_{n}=3\sum_{i=1}^{n}\frac{10^n-1}{9}$

Hence,

$S= 3\sum\frac{10^n}{9}-3\sum\frac{1}{9}$

$S= 3[\frac{10(1-10^n)}{(1-10)\times9}-(\frac{1}{9})\times n$

$= 3[10\times\frac{10^n-1}{81}-\frac{n}{9}]$

Solving this we get,

$S= [10\times \frac{10^n-1}{27}-\frac{n}{3}]$

Hence, the sum of first n terms of the series is,

$$= \frac{10^{n+1}-10-9n}{27}$$