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I'm trying to solve the following recurrence relation. I've gotten through two iterations, but I don't see any pattern. I would appreciate any help with this.

$T(1)=1$

$T(n)=6\ T(n/6)\ +\ 2n\ +\ 3$

I've manually ran through two iterations and found the following:

$T(n/6)= 36T(n/36)+14n+21$

$T(n/36)=216(n/216)+86n+129$

The pattern that I see is:

$T(n) = 6^k(n/(6^k))+ (I\ can't\ determine\ the\ pattern\ of\ these\ terms)$

Rusty
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    The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! – Raphael May 05 '16 at 11:35

2 Answers2

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It's probably easier solving this using the Master Theorem.

T(n) = 6T(n/6) + 2n + 3

enter image description here

So in this case A=6, B=6, and D=1 because f(n) = 2n + 3 and 2n^1 + 3 = 2n + 3

We of course assume the base case is a constant such that T(1) = C

So we can easily see that the answer for this is T ( n ) = Θ ( n log n ) , since A is equal to B to the power of D.

We can see that is true just by plugging in the values

= A = B^D

= 6 = 6^1

= 2 > 1 (which is always true)

This guy on YouTube has a very nice video explaining how to solve recurrences using the Master theorem, I will leave the link here.

tempmail
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Hint:

$$T(n)=6\ T(n/6)\ +\ 2n\ +\ 3$$

$$T(n)=6\ (6\ T(n/6^2)\ +\ 2(n/6)\ +\ 3)\ +\ 2n\ +\ 3$$

$$T(n)=6\ (6\ (6\ T(n/6^3)\ +\ 2(n/6^2)\ +\ 3)\ +\ 2(n/6)\ +\ 3)\ +\ 2n\ +\ 3$$

Distribute the 6's.

crackpotHouseplant
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    as a general rule you'll want to expand your stuff out like this so you can see the pattern more easily, rather than grouping over and again – crackpotHouseplant May 05 '16 at 03:03