Question

The sum of first $n$ odd natural numbers is

• A. $2n+1$
• B. $n^2$
• C. $n^2–1$
• D. $n^2+1$

Answer 1

The correct option is B

$n^2$

Explanation for Correct answer:

Arithmetic Progression

1. Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also called Arithmetic Sequence.
2. In AP, we will come across some main terms, which are denoted as:

a. First term ($a$)
b. Common difference ($d$)
c. nth Term ($a_n$)
d. Sum of the first $n$ terms ($S_n$)

Option (b): $n^2$

The Sum of an AP is: $S=\frac{n}{2}\times (2a+(n-1\left)d\right)$

The sum of first n odd natural numbers are given by $1,3,5,7,9\dots \dots (2n–1)$ and this forms an AP.

$a=1$

$d=2$

$t_n=(2n–1)$

Sum of first od numbers are calculated as follow:

$$\begin{gathered} S=\frac{n}{2}\times (2a+(n-1\left)d\right) \\ S=\frac{n}{2}\times (2+2n-2) \\ S=\frac{n}{2}\times 2n \\ S=n^2 \end{gathered}$$

Hence, Option (b) is the correct answer.

Explanation for incorrect answer:

Option (a): $2n+1$

Option (c): $n^2–1$

Option (d): $n^2+1$

As the sum of first $n$ odd natural numbers is $n^2$ as determined above.

Thus, option (a), (c) and (d) are incorrect answers.

Hence, option (b) is the correct answer.