Find the sum#8546

Find the sum (Editorial)

Compute the specified sum using a loop.

Let's calculate the specified sum mathematically:

\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{n \cdot ( n + 1 )} = (\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + ... + (\frac{1}{n} - \frac{1}{n + 1}) = 1 - \frac{1}{n + 1}

{1 \over 1 \cdot 2} + {1 \over 2 \cdot 3} + ... + {1 \over n \cdot (n + 1)} = \\ \left( {1 \over 1} - {1 \over 2} \right) + \left( {1 \over 2} - {1 \over 3} \right) + ... + \left( {1 \over n} - {1 \over n + 1} \right) = 1 - {1 \over n + 1}

Now we can use this formula to find the answer.

Algorithm implementation

Read the input value of $n$ .

scanf("%d",&n);

In the variable $i$ , iterate through the numbers $1, 2, 3, ..., n$ 1, 2, 3, ..., n. On each iteration, add the value of $1/ (i \cdot (i + 1))$ 1 / (i \cdot (i + 1)) to the resulting sum $s$ .

s = 0; i = 1;
while (i <= n) 
{
  s += 1.0 / i / (i + 1);
  i++;
}

Print the answer.

printf("%lf\n",s);

Algorithm implementation — for loop

Read the input value of $n$ .

scanf("%d",&n);

Compute the desired sum in the variable $s$ .

s = 0;

Using a for loop, compute the sum of $n$ terms. The $i$ -th term is equal to $1/ (i \cdot (i + 1))$ 1 / (i \cdot (i + 1)).

for (i = 1; i <= n; i++)
  s += 1.0 / (i * (i + 1));

Print the answer.

printf("%.6lf\n",s);

Algorithm implementation — formula

Read the input value of $n$ .

scanf("%d", &n);

Compute the answer using the formula and print it.

s = 1 - 1.0 / (n + 1);
printf("%.6lf\n", s);

Java implementation

import java.util.*;

class Main 
{
  public static void main(String[] args) 
  {
    Scanner con = new Scanner(System.in);
    int n = con.nextInt();

    double s = 0;
    for(int i = 1; i <= n; i++)
      s += 1.0 / (i * (i + 1));

    System.out.printf("%6f\n", s);
    con.close();
  }
}

Java

Python implementation

Read the input value of $n$ .

n = int(input())

Compute the desired sum in the variable $s$ .

s = 0;

Using a for loop, compute the sum of $n$ terms. The $i$ -th term is equal to $1/ (i \cdot (i + 1))$ 1 / (i \cdot (i + 1)).

for i in range(1, n+1):
  s += 1.0 / (i * (i + 1));

Print the answer.

print(s)