The document outlines a proof by induction for the inequality 2k! < 4^k(k!)^2, starting with the base case of n=1. It assumes the inequality holds for n=k and proves it for n=k+1, establishing the left side is less than 4^(k+1). The proof concludes that the inequality is confirmed for all integer values of n greater than or equal to 1.

1. Prove the following inequality by induction on n:
Solution
Let n =1
Then 2C1 = 2 < 4
Hence this is true for n =1
Let this be true for n =k
Or 2kCk < 4^k
i.e. 2k! < 4^k(k!)^2
Consider 2k+1 C (k+1)
= = <
=
For K+1>2
1/(k+1) <1/2
Hence we get for k +1, left side
< 4k+1
Thus proved by induction