For the series defined by a[0] = 0, a[i] = 2a[i-1] + i, WolframAlpha gives me the sum as 2^(n+1) - n - 2. (See http://www.wolframalpha.com/input/?i=sum+of+2%5E%28n-i%29+*+i%2C+i+%3D+1+to+n)
But I don't understand how it reaches the simplified form. Can anybody help?
Let my try. We have $$a_i + i + 2 = 2[a_{i-1} + (i-1) + 2].$$
Then, set $b_i = a_i + i+ 2$, we have $b_0 = 2$ and $b_i = 2b_{i-1}$. Then $b_n = 2^nb_0 = 2^{n+1}$. So, $a_n = 2^{n+1} - n -2$.