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Factorial

by Tamagna
(Kolkata)










































If m^2 < m! then prove that (m+1)^2 < (m+1)!.

Use mathematical induction

Induction assumption (induction hypothesis): n² < n! for n > 3.

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Mar 28, 2012
Factorial – Inductive Proof
by: Staff

Question:

by Tamagna
(Kolkata)

If m^2 < m! then prove that (m + 1)^2 < (m + 1)!.


Answer:

n² < n!

This is true for every natural number n > 3

n² < n!, n ∈ ℕ, n ≥ 4

P(n) is the statement n² < n!



Using mathematical induction:

Induction assumption (induction hypothesis): n² < n! for n > 3.


1. Verify that P(4) is true. Base Case. For n = 4

4² < 4!

16 < 24

Yes, n² < n! is true for n = 4.


2. Inductive Step.

Assume that for any given k > 3, k² < k! is true.

P(k) = k² < k!

?? P(k+1) = (k+1)² < (k+1)!
since k > 3,

1 < k - 1

1(k+1) < (k-1)(k+1)

k+1 < k2 - 1 < k2 < k!

k+1 < k!

(k+1)(k+1) < (k+1)k!

(k+1)² < (k+1)!




Thanks for writing.

Staff
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