Q115 / 180 · COMEDK 2024 (Memory Based) · Mathematics
$\sum_{k=1}^{20} k k$ !is equal to
- A21 !
- B20 ! + 1
- Correct answerC$21!-1$
- D$20!-1$
Solution
We have, $\sum_{k=1}^{20} k k$ !
$\begin{aligned}<br/>\sum_{k=1}^{20} k k! & =\sum_{k=1}^{20}(k+1-1) k!=\sum_{k=1}^{20}[(k+1) k!-k!] \\<br/>& =\sum_{k=1}^{20}(k+1)!-\sum_{k=1}^{20} k![\because(n+1) n!=(n+1)! \\<br/>& =2!-1!+3!-2!+21!-3!+\ldots+21!-20! \\<br/>& =21!-1<br/>\end{aligned}$
$\begin{aligned}<br/>\sum_{k=1}^{20} k k! & =\sum_{k=1}^{20}(k+1-1) k!=\sum_{k=1}^{20}[(k+1) k!-k!] \\<br/>& =\sum_{k=1}^{20}(k+1)!-\sum_{k=1}^{20} k![\because(n+1) n!=(n+1)! \\<br/>& =2!-1!+3!-2!+21!-3!+\ldots+21!-20! \\<br/>& =21!-1<br/>\end{aligned}$
Hence, the correct answer is option: 'C'.