Question

Prove by mathematical induction that all Furino-sequences are superincreasing

Answer 1

im not sure this is 100% correct, but this is what I have. prove that the base case (n=2) is true, which is easy. the assume that \[f_k >2+4+...+f_{k-1}\] for \[k \ge3\] (the inductive hypothesis).

Answer 2

then the inductive step is as follows (i think):
\[f_{k+1}>2+4+...+f_{k-1}+f_k\]
\[f_{k+1}>f_k+f_k\] (by the inductive hypothesis)
\[f_{k+1}\ge2*f_k\] (since \[f_{k+1}=r_{k+1}*f_k\] and \[r_{k+1}\] is 2 or 5)
and we're done. Again, I know there are errors in this proof, but maybe it'll help you :)

Answer 3

ty