Induction Question (1 Viewer)

[fuckit1991]

Hey I was wondering if anyone could help me out with this question...

show that (1/n!)<(1/2^(n-1))...red aloud says one over n factorial is less than one over 2 to the power of (n-1)...i haven't quite learnt the new system of posting maths eqns...anyhow any help would be greatly appreciated.

 

[3unitz]

true for n = 3

assume true for n = k:

S_k: 1/k! < 1/2^(k-1) (k > 3)

now consider n = k+1:

S_k+1: 1/(k+1)! < 1/2^k

(1/k!) . [1/(k+1)] < [1/2^(k-1)] . (1/2)

using the assumption that S_k is true, then S_k+1 is true under the condition that:

[1/(k+1)] < (1/2)

k > 1

since we're only considering k > 3 this is not a problem, and S_n is true by induction for all n > 3.