Question

The sum of the odd numbers between 0 and 50 is :
A. 525
B. 625
C. 425
D. 725

Answer 1

Hint: The first odd number between 0 and 50 is 1 and the last odd number between 0 and 50 is 49. All the odd numbers between 0 and 50 are in AP having a common difference of 2. To solve this question, we will directly use the formula to find the sum of AP, given by, ${{S}_{n}}=\dfrac{n}{2}.\left\{ 2a+\left( n-1 \right)d \right\}$, where n = number of terms, a = first term and d = common difference.

Complete step-by-step answer:
It is given in the question that we have to find the sum of the odd numbers between 0 and 50. So, if we find the smallest odd number between 0 and 50, then it is 1 and the greatest odd number between 0 and 50 is 49. So, we have an AP of odd numbers between 0 and 50 with a common difference of 2. We will first try to find the total number of terms in the AP using the formula, ${{T}_{n}}=a+\left( n-1 \right)d$, where ${{T}_{n}}$ is the last term of the AP, a is the first term and d is the common difference. So, apply the formula, we get,
$\begin{align}
  & 49=1+\left( n-1 \right)2 \\
 & \Rightarrow 49=1+2n-2 \\
 & \Rightarrow 49=2n-1 \\
 & \Rightarrow 2n=50 \\
 & \Rightarrow n=25 \\
\end{align}$
So, there are 25 terms in the given AP. Now, we can find the sum of the AP by suing the direct formula for the same, that is, ${{S}_{n}}=\dfrac{n}{2}.\left\{ 2a+\left( n-1 \right)d \right\}$, where n = number of terms, a = first term and d = common difference. So, we have n = 25, a = 1 and d = 2. So, applying these in the formula, we get,
$\begin{align}
  & {{S}_{n}}=\dfrac{25}{2}\left\{ 2\times 1+\left( 25-1 \right)2 \right\} \\
 & \Rightarrow {{S}_{n}}=\dfrac{25}{2}\left\{ 2+24\times 2 \right\} \\
 & \Rightarrow {{S}_{n}}=\dfrac{25}{2}\left\{ 2+48 \right\} \\
 & \Rightarrow {{S}_{n}}=\dfrac{25\times 50}{2} \\
 & \Rightarrow {{S}_{n}}=25\times 25 \\
 & \Rightarrow {{S}_{n}}=625 \\
\end{align}$
Hence, we get the sum of the odd numbers between 0 and 50 as 625.
Therefore, option B is the correct answer.

Note: Many students get confused while solving this question, whether 0 is an even number or an odd number. We must remember that 0 is neither an even nor an odd number. The smallest odd number is 1. If we consider 0 as an odd number for this question, we will get n = 26 and that will lead to an incorrect answer.