Question

Find the sum of all odd numbers between $0$ and $50$.

Answer 1

Step 1: Find the number of terms

The list of odd numbers between$0$ and$50$ are

$1,3,5,7,9,\dots .49.$

To find the sum of all the odd numbers between$0–50$ we use the theory of arithmetic progression.

The first term $a=1$

The common difference, $d=2$

Total number of terms,$n=?$

The $nth$ term, $a_n=49$

So,

$$\begin{gathered} a_n=a+(n-1)d \\ \Rightarrow 49=1+(n–1)\times 2 \\ \Rightarrow 49=1+2n–2 \\ \Rightarrow 49=2n-1 \\ \Rightarrow 50=2n \\ \Rightarrow n=25 \end{gathered}$$

Step 2: Find the sum of terms

The sum of all odd numbers between $0$ and $50$ can be calculated using the formula

$$\begin{gathered} S_n=\frac{n}{2}(a+a_n) \\ {S}_n=\frac{25}{2}(1+49) \\ {S}_n=\frac{25}{2}\times \left(50\right) \\ {S}_n=625 \end{gathered}$$

Hence, the sum of all odd numbers between $0$ and $50$ is $625$