Consider the recurrence relation

$\displaystyle t_{(2^k)} = 2^{k} * (t_{(2^{k-1}}) + 5$

$\displaystyle t_{(1)} = 1$

So far I have done this using repeated substitution (iteration).

I have no idea if what I am doing is correct, or if this is the proper way to go about it. Any help would be great.

Step 1:

$\displaystyle t_{(2^k)} = 2^{k} * (t_{(2^{k-1}}) + 5$

Step 2:

$\displaystyle t_{(2^k)} = 2^{k-0} * (2^{k-1} * (t_{(2^{k-2})})) + 5 + 5$

$\displaystyle t_{(2^k)} = 2^{2k-1} * (t_{(2^{k-2})}) + 5^2$

Step 3:

$\displaystyle t_{(2^k)} = 2^{2k-1} * (2^{k-2} * (t_{(2^{k-3})})) + 5 + 5 + 5$

$\displaystyle t_{(2^k)} = 2^{3k-3} * (t_{(2^{k-3})}) + 5^3$

Step 4:

$\displaystyle t_{(2^k)} = 2^{3k-3} * (2^{k-3} * (t_{(2^{k-4})})) + 5 + 5 + 5 + 5$

$\displaystyle t_{(2^k)} = 2^{4k-6} * (t_{(2^{k-4})}) + 5^4$