Question

What is the sum of all odd numbers between $ 0 $ and $ 100? $

Answer 1

Hint: First of all we will find the odd numbers between the given range. Odd numbers can be defined as the number which can not be divided by the number two. To find the sum of all the numbers between the range we will use the formula for the arithmetic progression.

Complete step by step solution:
The odd numbers between $ 0 $ and $ 100 $ are found by dividing the numbers by two and when the remainder comes one it is the odd number.
Therefore, odd numbers odd numbers between $ 0 $ and $ 100 $ are $ 1,3,5,........97,99 $
Here the sequence follows the pattern where the difference between the two terms remains the constant.
Here, first term $ a = 1 $
Common difference, $ d = 3 - 1 = 2 $
Last term, $ l = 99 $
To find the number of terms in the series consider the last term as the nth term.
 $ {a_n} = a + (n - 1)d $
Place the values in the above equation –
 $ 99 = 1 + (n - 1)2 $
Simplify the above expression –
 $ 99 = 1 + 2n - 2 $
Simplify –
 $
  99 = 2n - 1 \\
  99 + 1 = 2n \\
  100 = 2n \\
  n = \dfrac{{100}}{2} \\
  n = 50 \;
 $
Now, by using the standard formula for the sum of terms in the arithmetic progression is given by –
 $ {S_n} = \dfrac{n}{2}[a + l] $
Place the known terms in the above equation –
 $ {S_{50}} = \dfrac{{50}}{2}[1 + 99] $
Simplify the above expression, common factors from the numerator and the denominator cancel each other.
 $ {S_{50}} = 25[100] $
Simplify the above expression finding the product of the terms –
 $ {S_{50}} = 2500 $
This is the required solution.
So, the correct answer is “2500”.

Note: Allows cross check the sequence so formed from the given data and then identify the correct pattern of the sequence so formed. When there is a common difference between the terms then it is the arithmetic progression and when there is a common ratio between the terms then it is the geometric progression.