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# Find the sum of N terms of the series 0.1, 0.11, 0.111, …

• Last Updated : 16 Aug, 2022

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-

## C++

 // C++ program to implement // the above approach #include  using namespace std;   // Function to return sum of // N term of the series double findSum(int N) {     int a = pow(10, N);     return (double)(N * 9 * a - a + 1)            / (81 * a); }   // Driver Code int main() {     int N = 6;     cout << findSum(N); }

## Java

 // Java program for the above approach import java.io.*; import java.lang.*; import java.util.*;   class GFG {   // Function to return sum of // N term of the series static double findSum(double N) {     double a = Math.pow(10, N);     return (double)(N * 9 * a - a + 1)            / (81 * a); }   // Driver Code     public static void main (String[] args) {           double N = 6;         System.out.print(findSum(N));     } }   // This code is contributed by hrithikgarg03188.

## Python3

 # Python 3 program for the above approach   # Function to return sum of # N term of the series def findSum(N):     a = pow(10, N)     return (N * 9 * a - a + 1) / (81 * a)   # Driver Code if __name__ == "__main__":         # Value of N     N = 6        print(findSum(N))   # This code is contributed by Abhishek Thakur.

## C#

 // C# program to implement // the above approach using System; class GFG {     // Function to return sum of   // N term of the series   static double findSum(int N)   {     int a = (int)Math.Pow(10, N);     return (double)(N * 9 * a - a + 1)       / (81 * a);   }     // Driver Code   public static void Main()   {     int N = 6;     Console.Write(findSum(N));   } }   // This code is contributed by Samim Hossain Mondal.

## Javascript

 

Output

0.654321

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

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