Question

Find the sum of the following series:

S = 5+ 55 +555+ 5555+........ upto n terms. ​

Answer 1

Answer:

$\boxed{\sf \: 5 + 55 + 555 + 5555 + .. \: n \: terms = \: \dfrac{5}{9}\left[\dfrac{10( {10}^{n} - 1)}{9} - n \right] \: }$

Explanation:

Given series is

$\sf \: 5 + 55 + 555 + 5555 + ... \: n \: terms$

$\sf \: = \: 5(1 + 11 + 111 + 1111 + ... \: n \: terms)$

$\sf \: = \: \dfrac{5}{9} (9 + 99 + 999 + 9999 + ... \: n \: terms)$

$\sf \: = \: \dfrac{5}{9} [(10 - 1) + (100 - 1) + (1000 - 1) + ... \: n \: terms]$

$\sf \: = \: \dfrac{5}{9}[ 10 + 100 + 1000 + .. \: n \: terms) - (1 + 1 + 1 + .. \: n \: terms)]$

We know,

Sum of n terms of a GP series having first term a and common ratio r is given by

$\begin{gathered}\begin{gathered}\bf\: S_n = \begin{cases} &\sf{\dfrac{a( {r}^{n} - 1) }{r - 1} }, \: \: r \: \ne \: 1 \\ \\ &\sf{\qquad \: na, \: \: \: \: r = 1} \end{cases}\end{gathered}\end{gathered}$

So, using these results, we get

$\sf \: = \: \dfrac{5}{9}\left[\dfrac{10( {10}^{n} - 1)}{10 - 1} - n \times 1 \right]$

$\sf \: = \: \dfrac{5}{9}\left[\dfrac{10( {10}^{n} - 1)}{9} - n \right]$

Hence,

$\implies\sf \: 5 + 55 + 555 + 5555 + .. \: n \: terms = \: \dfrac{5}{9}\left[\dfrac{10( {10}^{n} - 1)}{9} - n \right]$

Answer 2

$\huge\pink{\mid{\underline{\overline {\tt Answer:-}} \mid}}$

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$c \: language$

#include <stdio.h>

#include <math.h>

int main()

{

int n, i, sum = 0;

printf("Enter the number of terms: ");

scanf("%d", &n);

for (i = 1; i <= n; i++)

{

sum += (int)(pow(10, i) - 1) / 9 * 5;

}

printf("The sum of the series is: %d", sum);

return 0;

}

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$java -$

import java.util.Scanner;

public class Main {

public static void main(String[] args) {

Scanner sc = new Scanner(System.in);

System.out.print("Enter the number of terms: ");

int n = sc.nextInt();

int sum = 0;

for (int i = 1; i <= n; i++) {

int num = Integer.parseInt("5" + String.format("%0" + (i - 1) + "d", 0));

sum += num;

}

System.out.println("Sum of the series: " + sum);

}

}

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$python$

def sum_series(n):

sum = 0

for i in range(1, n+1):

sum += int(str(5) * i)

return sum

n = int(input("Enter the number of terms: "))

print("The sum of the series upto", n, "terms is:", sum_series(n))

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$\boxed{Thanks}$

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${\red {I \: hope \: this \: may \: help \: you}}$