Iterative recurrence.. Iteration method Here is the Equation and how far I got into solving this problem using the iteration method:
$$T(1) = 8  \\
T(n) = 3T(n-1) - 15$$
Iterations:
$i=1, $ 
$$T(n) = 3(3T(n-2) - 15) -15$$
$i=2, $ 
$$ 3(3(3T(n-3) - 15) -15) - 15$$
$i=3,$
 $$3(3(3(3T(n-4) - 15) -15) - 15) - 15$$
$i=4,$
$$ 3(3(3(3(3T(n-5) - 15) -15) - 15) - 15) - 15$$
From the iteration pattern I found that
$$T(n) = 3^{i+1} \times T(n-(i+1)) - 15$$
At this point I need to find a summation for this recurrence and make it into closed form... I'm just not sure how.
Can someone explain or help guide me to solving this problem? 

It is easier to use generating functions. Define $g(z) = \sum_{n \ge 0} T(n + 1) z^n$. Write:
$$
T(n + 2) = 3 T(n + 1) - 15 \quad T(1) = 8
$$
Multiply by $z^n$ and add over $n \ge 0$:
$$
\frac{g(z) - T(1)}{z} = 3 g(z) - 15 \frac{1}{1 - z}
$$
From here:
$$
g(z) = \frac{8 - 23 z}{1 - 4 z + 3 z^2}
     = \frac{1}{2} \cdot \frac{1}{1 - 3 z} + \frac{15}{2} \cdot \frac{1}{1 - z}
$$
This gives:
$$
T(n) = \frac{1}{2} \cdot 3^{n + 1} + \frac{15}{2}
     = \frac{3^{n + 1} + 15}{2}
$$
