Find the sum of N terms of the series 0.1, 0.11, 0.111, ...

Given a positive integer, N. Find the sum of the first N term of the series-

 0.1, 0.11, 0.111, 0.1111, ....till N terms

Examples:

Input: N = 6
Output: 0.654321

Input: N = 1
Output: 0.1

Approach:

1st term = 0.1

2nd term = 0.11

3rd term = 0.111

4th term = 0.1111

.

.

Nth term = 1/9(1 - (1/10) ^ N)

The sequence is formed by using the following pattern. For any value N-

Derivation:

The following series of steps can be used to derive the formula to find the sum of N terms-

The series 0.1, 0.11, 0.111, ...till N terms can be written as

     -(1)

The series  is in G.P. with

First term a = 0.1  = 10^-1

Common Ratio r = 10^-1

Sum of G.P. for r<1 can be expressed as-

Substituting the values of a and r in the equation-

        -(2)

Substituting the equation (2) in (1), we get-

Illustration:

Input: N = 6
Output: 0.654321
Explanation:

Below is the implementation of the above approach-

0.654321

Time Complexity: O(logN) because it is using inbuilt pow function
Auxiliary Space: O(1)