Question 1: If $P (n)$ is the statement $n (n + 1)$ is even, then what is $P (3)$ ?

Given $P(n) = n(n+1)$ $\therefore P(3): 3 ( 3 + 1) = 12$ $\\$

Question 2: If $P (n)$ is the statement $n^3 + n$ is divisible by $3$, prove that $P(3)$ is true but $P (4)$ is not true.

Given $P(n) : n^3 + n$ is divisible by $3$ $P(3) : 3^3 + 3 = 37 + 30$  which is divisible by $3 \Rightarrow P(3)$ is true. $P(4): 4^3 + 4 = 64 + 4 = 67$ which is not divisible by $3 \Rightarrow P(4)$ is not true. $\\$

Question 3: If $P (n)$ is the statement $2^n \geq 3n$ , and, if $P(r)$ is true; prove that $P(r + 1)$ is true.

Given $P(n): 2^n \geq 3n$

Given $P(r)$ is true $\Rightarrow 2^r \geq 3r$ $\Rightarrow 2^{r+1} \geq 6r$

Since $3r \geq 3 \Rightarrow 3r+3r \geq 3+3r$ $\Rightarrow 2^{r+1} \geq 3(r+1)$ $\Rightarrow P(r+1)$ is true $\\$

Question 4: If $P (n)$ is the statement $n^2+n$ is even, and. if $P(r)$ is true, then $P (r + 1)$ is true.

Given $P(n) : n^2 + n$ is even

Given $P(r)$ is true $\Rightarrow P(r): r^2 + r$ is even $\Rightarrow r^2 + r = 2\lambda$ where $\lambda \in N$ $P(r+1): (r+1)^2 + ( r+1)$ $= r^2 + 2r + 1 + r + 1$ $= ( r^2 + r) + 2 ( r+1)$ $= 2 \lambda + 2 (r+1)$ $= 2 ( \lambda+r+1)$ $= 2 \mu$  where $\mu = \lambda+r+1$ $\Rightarrow P(r+1)$ is even $\Rightarrow P(r+1)$ is true $\\$

Question 5: Give an example of a statement $P (n)$ such that it is true for all $n \in N$. $P(n): 1 + 2 + 3 + \ldots + n =$ $\frac{n(n+1)}{2}$ is true for all $n \in N$ $\\$

Question 6: If $(n)$ is the statement $n^2-n +41$ is prime, prove that $P (1), P (2)$ and $P (3)$ are true. Prove also that $P (41)$ is not true.

Given $P(n): n^2-n +41$ is prime $P(1) : 1^2 - 1 + 41 = 41$ is a prime number. Therefore $P(1)$ is true. $P(2) : 2^2 - 2 + 41 = 43$ is a prime number. Therefore $P(2)$ is true. $P(3) : 3^2 - 3 + 41 = 47$ is a prime number. Therefore $P(3)$ is true. $P(41) : 41^2 - 41 + 41 = 41^2$ is not a prime number. Therefore $P(41)$ is not true. $\\$

Question 7: Give an example of a statement $P(n)$ which is true for all $n \geq 4$ but $P(1), P(2)$ and $P( 3)$ are not true. Justify your answer.

Let $P(n) : 3n < n!$ $P(1) : 3 \times 1 < 1!$ . Therefore $P(1)$ is not true. $P(2) : 3 \times 2 < 2!$ . Therefore $P(1)$ is not true. $P(3) : 3 \times 3 < 3!$ . Therefore $P(1)$ is not true. $P(4) : 3 \times 4 < 4!$ . Therefore $P(1)$ is true. $P(5) : 3 \times 5 < 5!$ . Therefore $P(1)$ is true. $\\$