lower and upper bounds

Question

i have the following recurrence relation:

T(n) = 2T(n/3) +5(n/6) + n

and can't quite figure out the right lower and upper bounds.
for the upper bound I did:

T(n) = 2T(n/3) +5T(n/3) +n = 7T(n/6) +n

which, according to the master theorem should be: n^log₆7

and for the lower bound i did:

T(n) = 2T(n/3) +5T(n/3) +n = 7T(n/3) +n

which, according to the master theorem should be: n^log₃7

however, when solving it that wat i got only partial score as "there are better bounds"
what an i doing wrong?

Answer 1

T(n)=2*T(n/3)+5*T(n/6)+n

Let
T(n)=c*n^k+o(n^k)

Then
c*n^k+o(n^k)=2*c*(n/3)^k+o((n/3)^k)+5*c*(n/6)^k+o((n/6)^k)+n  

c/6^k*(6^k-2^(k+1)-5)=o(n^k)

k=1.27635...

So, T(n) ≃ c*n^1.27635

T(n) ∈ Θ(n^1.27635)