The sequence s0,s1,s2... is defined by s0=1 and for all integers n>0, $s(n)=s(⌊n/2⌋)+s(⌊2n/5⌋) + n.$ Prove, using strong induction, that S(n) > 4n for all integers n>=3.
To my knowledge, I only have to prove a single base case n=3. What I am not sure about is how to do the induction step, specifically the following:
Let k be an integer $k >= ? $. Suppose that for all integers $m$, where $0<=m<k$....
I am not sure how to determine what value '?' should be for k, and also what dictates why the lower range for m is 0 and not something else?
Here are the first 8 values of the sequence for reference:
S1= S0 + S0 + 1 = 3
S2= S1 + S1 + 2 = 8
S3= S1 + S2 + 3 = 14
S4= S2 + S2 + 4 = 20
S5= S2 + S3 + 5 = 27
S6= S3 + S4 + 6 = 40
S7= S3 + S4 + 7 = 41
S8= S4 + S5 + 8 = 55