Question

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

• A. all $nϵN$
• B. all n > 1
• C. all n > 1 and $x\ne 0$
• D. None of these

Answer 1

The correct option is C all n > 1 and $x\ne 0$

If $x>1$ then the statement

$P\left(n\right):(1+x{)}^n>1+nx$

$P\left(1\right):(1+x{)}^1>1+x$, which is false

$P\left(2\right):(1+x{)}^2>1+2x$, which is true if $x\ne 0$

Let $P\left(k\right):(1+x{)}^k>1+kx$ is true

$\Rightarrow (1+x)(1+x{)}^k>(1+x\left)\right(1+kx)$

$\Rightarrow (1+x{)}^{k+1}>(1+x\left)\right(1+kx)=1+(k+1)x+k{x}^2$

$\Rightarrow (1+x{)}^{k+1}>1+(k+1)x\because k{x}^2>0$

Hence $P\left(n\right)$ is true

so $P\left(n\right)$ is true for all $n>1$ and $x\ne 0$