Question

Find the sum of the first $25$ odd natural numbers.

Answer 1

Step 1: Define the problem

The sequence of odd numbers is in the form of an arithmetic progression
Arithmetic progression is a sequence where each term $a_{\mathrm{n}}$ is in the form of,

$a_{\mathrm{n}}=a+(n-1)d$

where, $a$ is the first term of the series, $d$ is the difference between any $n$^th and $(n-1)$^th term.

Odd natural numbers start with $1$ and increase by $2$ every succeeding term (i.e., their common difference).

Thus, $a=1$, $d=2$ and $n=25$.

Step 2: Apply the formula for the sum of AP

The sum ($S_n$) of an arithmetic progression is given as,

$S_n=\frac{n}{2}\left[2a+(n-1)d\right]$

Thus, the sum of first $25$ odd numbers is,

$\begin{array}{rcl}{S}_n& =& \frac{25}{2}\left[2\times 1+(25-1)·2\right]\\ & =& 625\end{array}$

Thus, the sum of the first $25$ odd natural numbers is $625$.