Question

\( \sum_{n=1}^{m} n \cdot n! \) is equal to

• A. \( m! - 1 \)
• B. \( (m - 1)! - 1 \)
• C. \( (m + 1)! - 1 \)
• D. \( m! (m - 1)! \)

Hint:

Memorize identities involving summations with factorials: they simplify pattern recognition.

Answer 1

The Correct Option is C

We are given: \[ \sum_{n=1}^{m} n \cdot n! = 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \ldots + m \cdot m! \] Observe the pattern:
Let’s take few terms:
\( 1 \cdot 1! = 1 \), \( 2 \cdot 2! = 4 \), \( 3 \cdot 3! = 18 \), etc.
Total: \( 1 + 4 + 18 = 23 \), and \( 4! = 24 \Rightarrow 24 - 1 = 23 \)
General Formula: \[ \sum_{n=1}^{m} n \cdot n! = (m+1)! - 1 \]