Question

For all real numbers $a,b$ and positive integer $n$ prove that:
$(a+b{)}^n{=}^n{C}_0{a}^n{+}^n{C}_1{a}^{n-1}b{+}^n{C}_1{a}^{n-2}{b}^2+..........{+}^n{C}_n{b}^n$.

Answer 1

Given: $\begin{array}{c}{\left(a+b\right)}^n{=}^n{c}_0{a}^n+{}^n{c}_1{a}^{n-1}b{+}^n{c}_2{a}^{n-2}{b}^2+.......{+}^n{c}_n{b}^n\end{array}$........(1)
Let us prove above result by principles of mathematical induction as follows:
Step(i): For $n=1$,
LHS$={\left(a+b\right)}^1$
$=(a+b)$
R.H.S${=}^1{c}_0{a}^1+{}^1{c}_1{a}^{1-1}b$
$=a+a^{0}b$
$=a+\left(1\right)b$
$=a+b$
Thus, LHS=RHS.
Therefore, the result is true for $n=1$.
Step(ii): Assume the result is true for $n=k$.
i.e., $\begin{array}{c}{\left(a+b\right)}^k{=}^k{c}_0{a}^k+{}^k{c}_1{a}^{k-1}b{+}^k{c}_2{a}^{k-2}{b}^2+.......{+}^k{c}_k{b}^k\end{array}$..........(2)
Step(iii): We shall prove that the result is true for $n=k+1$.
i.e., we need to prove $\begin{array}{c}{\left(a+b\right)}^{k+1}{=}^{k+1}{c}_0{a}^{k+1}+{}^{k+1}{c}_1{a}^{k}b+{}^{k+1}{c}_2{a}^{k-1}{b}^2+.......{+}^{k+1}{c}_{k+1}{b}^{k+1}\end{array}$ ........(3)
Consider LHS$={\left(a+b\right)}^{k+1}$
$=\left(a+b\right){\left(a+b\right)}^k$
$=\left(a+b\right)\times$$\left(\begin{array}{c}{}^k{c}_0{a}^k+{}^k{c}_1{a}^{k-1}b{+}^k{c}_2{a}^{k-2}{b}^2+.......{+}^k{c}_k{b}^k\end{array}\right)$ [From equation (2)]
$=\begin{array}{c}{}^k{c}_0{a}^{k+1}+{}^k{c}_1{a}^{k}b{+}^k{c}_2{a}^{k-1}{b}^2+.......{+}^k{c}_{k}a{b}^k\end{array}$$+\begin{array}{c}{}^k{c}_0{a}^{k}b+{}^k{c}_1{a}^{k-1}{b}^2{+}^k{c}_2{a}^{k-2}{b}^3+.......{+}^k{c}_k{b}^{k+1}\end{array}$
${=}^k{c}_0{a}^{k+1}+{(}^k{c}_1{+}^k{c}_0\left)\right(a^{k}b)+{(}^k{c}_2{+}^k{c}_1)\left(a^{k-1}{b}^2\right)+.......+{(}^k{c}_k{+}^k{c}_{k-1})a{b}^k{+}^k{c}_k{b}^{k+1}$
${=}^{k+1}{c}_0{a}^{k+1}{+}^{k+1}{c}_1\left(a^{k}b\right){+}^{k+1}{c}_2\left(a^{k-1}{b}^2\right)+.......{+}^{k+1}{c}_{k}a{b}^k{+}^{k+1}{c}_{k+1}{b}^{k+1}$ [${\because }^{k+1}{c}_0=1,{(}^n{c}_r{+}^n{c}_{r-1}){=}^{n+1}{c}_r{,}^{k+1}{c}_{k+1}=1$]
$=R.H.S$
Therefore, the result is true for $n=k+1$.
Hence, by principle of mathematical induction, the given results is true for all positive integers.