Question

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

Answer 1

Step 1: Defining the problem

The odd numbers between $0$ and $50$ are $1,3,5,...,49$

Therefore, this is an arithmetic progression with the first term, $a=1$, $d=2$ and $a_n=49$

Step 2: Finding number of terms, $n$

An arithmetic progression is defined as, $a_n=a+(n-1)d$

Thus,

$$\begin{gathered} 49=1+(n-1)·2 \\ \Rightarrow n-1=\frac{48}{2} \\ \Rightarrow n=25 \end{gathered}$$

Therefore, number of term, $n=25$

Step 3: Finding the sum of the series

Sum of an arithmetic progression ($S_n$) is given by,

$$\begin{gathered} S_n=\frac{n}{2}\left[2a+(n-1)d\right] \\ =\frac{25}{2}\left[2\times 1+(25-1)·2\right] \\ =625 \end{gathered}$$

Therefore the sum of the odd numbers between $0$ and $50$ is $625$