Question

If \(S(n):2n<n!\), \(n\in\mathbb N\), then \(S(n+1)\) is true for all \(n\ge\):

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

In induction problems, always verify the first value where the statement actually becomes true before applying the induction step.

Answer 1

The Correct Option is C

Concept: Use mathematical induction.

Step 1: Assume \(S(n)\) is true.
\[ 2n<n! \] for some \(n\).

Step 2: Check \(S(n+1)\).
We need \[ 2(n+1)<(n+1)! \] Since \[ 2n<n!, \] multiplying by \((n+1)\), \[ 2n(n+1)<(n+1)!. \] Thus it is sufficient that \[ 2(n+1)\le 2n(n+1). \] Cancelling \(2(n+1)\), \[ 1\le n. \] But the induction starts only when the base case holds.

Step 3: Check smallest valid value.
For \(n=2\), \[ 2n=4,\qquad n!=2 \] false. For \(n=3\), \[ 2n=6,\qquad n!=6 \] false. For \(n=4\), \[ 8<24 \] true. Therefore induction works from \[ n\ge 3. \] \[ \boxed{\text{Correct Option (3)}} \]