Question

If $P(n): 1 + 3 + 5 + \cdots + (2n-1) = n^2$ is

• A. true for all $n \in \mathbb{N}$
• B. true for $n>1$
• C. true for no $n$
• D. None of these

Hint:

The sum of the first $n$ odd natural numbers always equals $n^2$. This is a standard result worth memorising.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a classic result proved by mathematical induction or by the formula for the sum of an arithmetic progression.
Step 2: Detailed Explanation:
The sum of the first $n$ odd numbers is an AP with first term 1 and common difference 2.
$S_n = \dfrac{n}{2}[2(1) + (n-1)(2)] = \dfrac{n}{2}(2n) = n^2$.
This holds for every natural number $n$.
Step 3: Final Answer:
$P(n)$ is true for all $n \in \mathbb{N}$.