Question

The sum of first $n$ odd natural numbers is $n^2$. State whether the statement is true (T) or false (F).

• A. True
• B. False

Answer 1

The correct option is A True

Step 1: Natural numbers and Odd numbers

Natural numbers

Natural numbers are a part of the number system which includes all the positive integers from $1$ till infinity and are also used for counting purpose.

Odd numbers

Odd numbers are the numbers that cannot be divided by $2$ evenly.

We have to find sum of the series : $1+3+5+...tonterms$
Clearly, terms of the given series form an AP with first term $a=1$ and common difference $d=2.$ Therefore,

$$\begin{gathered} {\mathbf{}{\mathbf{S}}_{\mathbf{n}}\mathbf{}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{[}\mathbf{2}\mathbf{a}\mathbf{+}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathbf{d}\mathbf{\right]} \\ \Rightarrow S_n=\frac{n}{2}[2\times 1+(n-1)\times 2] \\ \Rightarrow S_n=\frac{n}{2}[2+2n-2] \\ \Rightarrow S_n=n^2}^{} \end{gathered}$$

Therefore, the given statement is true.