Can you use mathematical induction to prove that #5^n < n!# for all #n in ZZ^+, n>=12#?
1 Answer
Feb 3, 2017
See explanation...
Explanation:
Proposition
Let
#5^color(blue)(n) < color(blue)(n)!#
Base case
#5^color(blue)(12) = 244140625 < 479001600 = color(blue)(12)!#
Induction step
Suppose
#5^(color(blue)(n)) < color(blue)(n)!#
Then
#5^(color(blue)(n+1)) = 5*5^n < 5*n! < (n+1)*n! = (color(blue)(n+1))!#
Conclusion
Having shown:
#{ (P(color(blue)(12))), (P(color(blue)(n)) => P(color(blue)(n+1)) " for " n >= 12) :}#
We can deduce:
#P(n)# for all#n >= 12#