Q. If $(n):1 + 3 + 5 + ... + (2n - 1) = n^{2}$ is
A) True for all $n \in N$
B) True for $n > 1$
C) True for no $n$
D) None of these
Explanation
(a)
Given, $(n):1 + 3 + 5 + ... + (2n - 1) = n^{2}$
$P(1):1 = 1$(true)
Let $P(k) = 1 + 3 + 5 + ... + (2k - 1) = k^{2}$
$\therefore P(k + 1) = 1 + 3 + 5 + ... + (2k - 1) + 2k + 1$
$= k^{2} + 2k + 1 = (k + 1)^{2}$
So, it holds for all $n$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.