prove that : (1+x)(to power n ) > 1+nx ; for n>1 , x >-1

${\left(1+x\right)}^n>1+\mathrm{nx}$$\mathrm{Prove} \mathrm{by} \mathrm{Mathematical} \mathrm{induction}$$\mathrm{For} n=1, {\left(1+x\right)}^1=1+x$$\mathrm{Asume} \mathrm{the} \mathrm{result} \mathrm{for} n =k,$${\left(1+x\right)}^k>1+\mathrm{kx} -----------\left(1\right)$$T.P.T \mathrm{the} \mathrm{reult} \mathrm{is} \mathrm{true} \mathrm{for} n = k+1,$$i.e T.P.T {\left(1+x\right)}^{k+1}>1+(k+1)x$$\mathrm{Now} {\left(1+x\right)}^{k+1}={\left(1+x\right)}^k\left(1+x\right)$$>(1+\mathrm{kx})(1+x) \mathrm{by} \left(1\right)$$= 1+(k+1)x+{\mathrm{kx}}^2>1+(k+1)x \mathrm{as} {\mathrm{kx}}^2>0$$\mathrm{Hence} \mathrm{proved} \mathrm{by} \mathrm{induction}$">