If you have transcribed this correctly then you are right and the book is wrong.
I am trying to find the sum of a sequence, the problem is already solved in the book but not in a detailed way. While I understand the sum calculation process I don't understand the simplification result. I have marked in red the difference in the solution. In particular it says:
(na/2) - (a/2)*(g^1 + g^2 + ... + g^n) + (nb/2) + (b/2)*(g^1 + g^2+ ... + g^n) =
=(n/2)(a+b) + [(1-g^n)/(1-g)]*[(b-a)/2]
My result is this:
=(n/2)(a+b) + [(g-g^n+1)/(1-g)]*[(b-a)/2]
I thought it could be some error, but there is the same result in an older version of the book. It might be something easy but still I cannot configure it out.
Thank you all in advance,
Thank you very much for your kind a quick reply! It's of help to have a second opinion.
I double checked again that I have write correctly the book's solution here...I worked on it a lot and I am coming to same conclusion as you did:
The particular section, how solved it (very detailed):
(g^1 + g^2 + ... + g^n) =
[(...)*(1-g)] / [1-g] =
[(...)(1) + (...)(-g)] / [...] =
[(...) + (-1)(g)(...)] / [...] =
[(...) + (-1)(g^2 + g^2 +...+g^n + g^n+1)] / [...] =
[(...) + (-g^2 - g^3 - ... -g^n - g^n+1)] / [...]=
[g^1 + g^2 + ... + g^n -g^2 -g^3-...-g^n -g^n+1] / [...]=
[g^1 -g^n+1] / [1-g] or [g -g^n+1] / [1-g] .
Thank you Plato,
[If you haven't done already I would suggest you read the Socrates apology saved-written by Plato! Socrates reasoning, the way he acts in the court is exceptional.]