Introduction to P.M.I
Question

# Let $x>-1$, then statement $P\left(n\right):\left(1+x{\right)}^{n}>1+nx$ is true for

Moderate
Solution

## For n=2,$P\left(2\right):\left(1+x{\right)}^{2}=1+2x+{x}^{2}>1+2x$,as $x\ne 0$.Assume that$P\left(k\right):\left(1+x{\right)}^{k}>1+kx$                            (1)for some $k\in \mathbf{N},k>1$As $x>-1$, multiplying both the sides of (1) by $1+x$, we get$\begin{array}{l}\left(1+x{\right)}^{k+1}>\left(1+kx\right)\left(1+x\right)\\ =1+\left(k+1\right)x+k{x}^{2}\\ >1+\left(k+1\right)x\end{array}$Thus, $P\left(k+1\right)$ is true.By the principle of mathematical induction is true forall $n>1$ provided $x\ne 0$.

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