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.

Give an example of a statement $P(n)$ which is true for all $n$. Justify your answer.

$4^n-1$ is divisible by 3, for each natural number n.

$2^{3 n}-1$ is divisible by 7 , for all natural numbers $n$.

$n^3-7 n+3$ is divisible by 3 , for all natural numbers $n$.

$3^{2 n}-1$ is divisible by 8, for all natural numbers $n$.

For any natural numbers $n, 7^n-2^n$ is divisible by 5.

For any natural numbers $n, x^n-y^n$ is divisible by $x-y$, where $x$ and $y$ are any integers with $x \neq y$.

$n^3-n$ is divisible by 6, for each natural number $n\ge2$.

$n\left(n^2+5\right)$ is divisible by 6 , for each natural number $n$.

$n^2<2^n$, for all natural numbers $n \geq 5$.

$122 n<(n+2)$ ! for all natural numbers $n$.

$\sqrt{n}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\supset+\frac{1}{\sqrt{n}}$, for all natural numbers $n \geq 2$.

$142+4+6+\supset+2 n=n^2+n$, for all natural numbers $n$.

$1+2+2^2+\supset+2^n=2^{n+1}-1$ for all natural numbers $n$.

$1+5+9+\supset+(4 n-3)=n(2 n-1)$, for all natural numbers $n$.

A sequence $a_1, a_2, a_3, \ldots$ is defined by letting $a_1=3$ and $a_k=7 a_{k-1}$, for all natural numbers $k \geq 2$. Show that $a_n=3 \cdot 7^{n-1}$ for all natural numbers.

A sequence $b_0, b_1, b_2, \ldots$ is defined by letting $b_0=5$ and $b_k=4+b_{k-1}$, for all natural numbers $k$. Show that $b_n=5+4 n$, for all natural number $n$ using mathematical induction.

A sequence $d_1, d_2, d_3, \ldots$ is defined by letting $d_1=2$ and $d_k=\frac{d_{k-1}}{k}$, for all natural numbers, $k \geq 2$. Show that $d_n=\frac{2}{n!}$, for all $n \in N$.

Prove that for all $n \in N$ $$ \begin{aligned} \cos \alpha+ & \cos (\alpha+\beta)+\cos (\alpha+2 \beta)+\supset+\cos [\alpha+(n-1) \beta] \\ = & \frac{\cos \left[\alpha+\left(\frac{n-1}{2}\right) \beta\right] \sin \left(\frac{n \beta}{2}\right)}{\sin \frac{\beta}{2}} \end{aligned}$$

Prove that $\cos \theta \cos 2 \theta \cos 2^2 \theta D \cos 2^{n-1} \theta=\frac{\sin 2^n \theta}{2^n \sin \theta}, \forall n \in N$

$$ \begin{aligned} & \text { Prove that, } \sin \theta+\sin 2 \theta+\sin 3 \theta+\supset+\sin n \theta=\frac{\frac{\sin n \theta}{2} \sin \frac{(n+1)}{2} \theta}{\sin \frac{\theta}{2}} \text {, } \\ & \text { for all } n \in N \text {. } \end{aligned} $$

Show that $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7 n}{15}$ is a natural number, for all $n \in N$.

Prove that $\frac{1}{n+1}+\frac{1}{n+2}+\supset+\frac{1}{2 n}>\frac{13}{24}$, for all natural numbers $n>1$.

Prove that number of subsets of a set containing $n$ distinct elements is $2^n$, for all $n \in N$.

If $10^n+3 \cdot 4^{n+2}+k$ is divisible by 9 , for all $n \in N$, then the least positive integral value of $k$ is

If $x^n-1$ is divisible by $x-k$, then the least positive integral value of $k$ is

For all $n \in N, 3 \cdot 5^{2 n+1}+2^{3 n+1}$ is divisible by

If $P(n): 2 n < n!, n \in N$, then $P(n)$ is true for all $n \geq$ ........... .

Let $P(n)$ be a statement and let $P(k) \Rightarrow P(k+1)$, for some natural number $k$, then $P(n)$ is true for all $n \in N$.