In a single dimension world, a girl is standing at the point $x = 0$ . Her boyfriend is standing at some point $x = x_0$ . He is drunk. So, he either takes a step towards her girlfriend or a step away,with equal probability. But he can’t go beyond $x = n$ ,because of a wall placed there, So whenever he reaches $x = n$ ,he has to go back to $x = ( n - 1 )$ . Let the expected number of steps for him to reach his girlfriend be $F ( x_0 , n )$ where $x_0$ is his starting point.

Find the value of $\sum_{i=1}^{N} F (i, N)$ for $N = 10^7$