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I've been stuck for hours trying to solve the recurrence $T(n) = 7T(n/3) + n^2 + 2n$ by using case 3 of the master theorem.

I've done a good chunk of the proof, but currently stuck attempting to solve the regularity property $af(n/b) \leq cf(n)$.

I don't think this fits the case 3 requirements, but my best shot has been $(7n^2 / 9) + (14n / 3) \leq (7/9)n^2 + 2(14/3)n$. This is true for all $n$, but I'm using 2 constants. What should I do to find the sole constant $c$ (where $c$ is $0 < c < 1$) which would satisfy case 3?

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    $\begingroup$ Just ignore the $+ 2n$, and you have $a=7$, $b=3$, and $d=2$. $\endgroup$
    – Pål GD
    Commented Oct 5, 2023 at 13:15

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