R D Sharma Solutions for Chapter: Mathematical Induction, Exercise 1: EXERCISE 12.1

If $P \left(n\right)$ is the statement $“n (n+1)$ is even”, then what is $P \left(3\right)$?

If $P \left(n\right)$ is the statement $“n^3+n$ is divisible by $3”,$ prove that $P \left(3\right)$ is true but $P \left(4\right)$ is not true.

If $P\left(n\right)$ is the statement ${}^{\text{'}\text{'}}2^n\ge 3n^{\text{'}\text{'}},$ and if $P\left(r\right)$ is true, prove that $P(r+1)$ is true.

 If $P \left(n\right)$ is the statement $“n^2+n”$ is even$”,$ and if $P \left(r\right)$ is true, then $P (r+1)$ is true

Given an example of a statement $P \left(n\right)$ such that it is true for all $n\in N$.

 If $P \left(n\right)$ is the statement $“n^2–n+41$ is prime$”,$ prove that $P \left(1\right), P \left(2\right)$ and $P \left(3\right)$ are true. Prove also that $P \left(41\right)$ is not true.

Give an example of a statement $P\left(n\right)$ which is true for all $n\ge 4$ but $P\left(1\right), P\left(2\right)$ and $P\left(3\right)$ are not true. Justify your answer.