Hi guys. I am going through an example in my book, and I am stuck on the last step concerning the initial condition.

The coefficient derived from the initial condition is:

and the linear sum of the particular solutions is:

I think then

is substituted into the equation

above to get:

I am having difficulty working out the algebra on this one. Can anyone show it to me step by step. I can see that there is a factor (2n-1) but I don't know where they got that from. Can anyone explain? I am finding this really difficult to work out.