Question

Find the sum of the first $20$ odd natural numbers by AP method

Answer 1

Step 1: Defining the problem

The sequence of odd numbers is in the form of an arithmetic progression
Arithmetic progression are sequences where each term $a_n$ is in the form of,

$a_n=a+(n-1)d$

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 $n=20$.

Step 2: Apply the formula for 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 $20$ odd numbers is,

$$\begin{gathered} S_n=\frac{20}{2}\left[2\times 1+(20-1)·2\right] \\ =400 \end{gathered}$$

Thus, the sum of the first $20$ odd natural numbers is $400$.