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.

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