Q71 / 180 · COMEDK 2017 · Mathematics

The statement $P(n): ' 1 \times 1 !+2 \times 2 !+3 \times 3$ ! $+\ldots+n \times n !=(n+1) !-1$ is

  • A
    true for all $n>1$
  • B
    not true for any $n$
  • Correct answer
    C
    true for all $n \in N$
  • D
    None of these

Solution

$P(1): ' 1 \times 1 !=(1+1) !-1$ ' is true. Let $P(k):{ }^{\prime \prime} 1 \times 1 !+2 \times 2 !+3 \times 3 !$ $+\ldots+k \times k !=(k+1) !-1^{\prime \prime}$ is true. Now, $P(k+1): 1 \times 1 !+2 \times 2 !+\ldots$ $\begin{array}{r}+k+k !+(k+1) \times(k+1) !\end{array}$ $=(k+1) !-1+(k+1)(k+1) !$ $=(k+1) !+(k+1) \times(k+1) !-1$ $=(k+1) !(1+k+1)-1=(k+2) !-1$ i.e., $P(k)$ is true $\Rightarrow P(k+1)$ is true

Hence, the correct answer is option: 'C'.

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