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I was asked the following question.

let $a_{1},a_{2},....,a_{k}$ be positive real numbers whose sum is strictly less than 1. Consider a function f:N->$R^{+}$ such that

$f(n)<=f(a_{1}n)+f(a_{2}n)+....+f(a_{k}n)$+cn

for positive c and all sufficiently large sn. Prove by constructive induction that $f(n) ∈ O(n)$.

The problem is I don't know how to start while not having a condition for f(0) for example. I thought about adding another variable d(n)>=f(n) but I still don't know how to begin my induction.

Zok
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  • In general $a_1 n$ won't be in $N$. How is $f(a_1 n)$ defined? – btilly Feb 15 '17 at 01:42
  • "Constructive induction" is a different thing than plain induction is - Googling it gives some good demonstrations. Also, as btilly said, $a_in$ won't be an integer for most $n$ (for example, if $n$ is prime and $a_i$ is not an integer multiple of $1/n$). Are you missing some part of the problem definition? – Reese Johnston Feb 15 '17 at 01:46

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I proved this in my answer to my own question here:

If $T(n) = un + \sum_i T(\lfloor r_i n \rfloor) $, show that $T(n) = \Theta(n)$

The generalized induction comes in because the values at all the $a_i n$ are used.

marty cohen
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  • Thank you for your kind answer. That helped me. But I have a question, does that necessarly work if we had the sum of $R_{i}$= 1 and not <1 ? – Zok Feb 15 '17 at 17:23
  • I don't think so. The slightly different recursion $t(n)=t(n-1)+n$ has $t(n)\approx n^2/2$. – marty cohen Feb 15 '17 at 17:28