State and prove binomial theorem 
 

Let n be a positive integer and x, a be real numbers, then 

$$\begin{gathered} {(x+a)}^n{=}^n{C}_0.x^n{a}^0{+}^n{C}_1.x^{n-1}{a}^1{+}^n{C}_n.x^{n-2}{a}^2+\mathrm{.........}{+}^n{C}_r.x^{n-r}{a}^r+\mathrm{.....} \\ {+}^n{C}_n.x^0{a}^n \end{gathered}$$

We prove this theorem by using the principle of mathematical induction (on n). 
When 
$n=1,{(x+a)}^n={(x+a)}^1=x+a{=}^1{C}_0{x}^1{a}^0{+}^1{C}_1{x}^0{a}^1$

Thus the theorem is true for n = 1
Assume that the theorem is true for n = k$\ge$1
(where k is a positive integer). That is 

$$\begin{gathered} {(x+a)}^k{=}^k{C}_0.x^k{a}^0{+}^k{C}_1.x^{k-1}{a}^1{+}^k{C}_2.x^{k-2}{a}^2+\mathrm{.........}{+}^k{C}_r.x^{k-r}{a}^r+\mathrm{.....} \\ {+}^k{C}_k.x^0{a}^k \end{gathered}$$

Now we prove that the theorem is true when n = k + 1 also 

${(x+a)}^{k+1}=(x+a){(x+a)}^k$

$=(x+a)[k_{C_0}.x^k{a}^0+k_{C_1}.x^{k-1}{a}^1+k_{C_2}.x^{k-2}{a}^2+\mathrm{.........}+k_{C_r}.x^{k-r}{a}^r+\mathrm{.....}+{k_C}_{{ }_k}.x^0{a}^k]$$=k_{C_0}{x}^{k+1}+k_{C_1}{x}^k{a}^1+k_{C_2}{x}^{k-1}{a}^2+\mathrm{.........}+k_{C_r}{x}^{k-r+1}{a}^r+\mathrm{.....}+{k_C}_{{ }_k}.x^1{a}^k$

$=k_{C_0}{x}^k.a^1+k_{C_1}{x}^{k-1}{a}^2+k_{C_2}{x}^{k-2}{a}^3+\mathrm{.........}+k_{C_r}{x}^{k-r}{a}^{r+1}+\mathrm{.....}+{k_C}_{{ }_k}{a}^{k+1}$

${=}^k{C}_0.x^{k+1}.a^0+{(}^k{C}_1{+}^k{C}_0).x^k.a^1+{(}^k{C}_2{+}^k{C}_1).x^{k-1}.a^2$$+\mathrm{........}+{(}^k{C}_r{+}^k{C}_{r-1}).x^{k-r+1}.a^r+\mathrm{......}+{(}^k{C}_k{+}^k{C}_{k-1}).x^1.a^k+$${ }^k{C}_k.x^0.a^{k+1}$

Since ${ }^k{C}_0=1{=}^{k+1}{C}_0{,}^k{C}_r{+}^k{C}_{r-1}{=}^{\left(k+1\right)}{C}_r$

for $1\le r\le k{ }^k{C}_k=1{=}^{\left(k+1\right)}{C}_{\left(k+1\right)}$

$$\begin{gathered} {\left(x+a\right)}^{k+1}{=}^{\left(k+1\right)}{C}_0.x^{k+1}.a^0{+}^{\left(k+1\right)}{C}_1.x^k.a^1{+}^{\left(k+1\right)}{C}_2.x^{k-1}.a^2+ \\ .....{.}^{\left(k+1\right)}{C}_r.x^{k-r+1}.a^r+...{+}^{\left(k+1\right)}{C}_k.x^1.a^k{+}^{k+1}{C}_{k+1}.x^0.a^{k+1} \end{gathered}$$

Therefore the theorem is true for n = k +1 Hence, by mathematical induction, it follows
that the theorem is true of all positive integer n: