Question

The statement P(n) $"1\times 1!+2\times 2!+3\times 3!+...+n\times n!=(n+1)!1"$ is

• A. True for all n > 1
• B. Not true for any n
• C. True for all $nϵN$
• D. None of these

Answer 1

The correct option is C True for all $nϵN$

$P\left(1\right):1\times 1=(1+1)=(1+1)!-1$ is true

Let $p\left(m\right)$ be true

$\Rightarrow 1\times 1!+2\times 2!+3\times 3!+...+m\times m!=(m+1)!-1$

$\Rightarrow 1\times 1+2\times 2!+3\times 3!+...+m\times m!+(m+1)\times (m+1)!$

$=(m+1)!-1(m+2)(m+1)!=(m+1)!(m+2)-1$

$=(m+2)!-1$

$\Rightarrow p(m+1)$ is also true

$\therefore P\left(n\right)$ is true for each $n$