How can I solve this recurrence relation

Question

I have a recurrence relation:

I transformed it into

How can I prove T(n) is belong to BigOmega(2^n)

Answer 1

So let's look at T(n) = 2 * T(n - 1) + T(n/2) + 1/2 .

Let us compare this to the recurrence relation where the smaller terms are stripped:

S(n) = 2 * T(n - 1)

We can obviously see that

T(n) = Ω(S(n))

so it only remains to show that S(n) = Ω(2 ⁿ ) .

Let us expand S(n) :

S(n) = 2 * T(n - 1)

= 2 * 2 * T(n - 2)

= 2 * 2 * 2 * T(n - 3)

We notice that we obtain 2 * 2 *... * 2 = 2 ⁿ .

Therefore, S(n) = Ω(2 ⁿ ) and thus T(n) = Ω(S(n)) = Ω(2 ⁿ ) . □