Hi, I need help with this question. We have to add first n terms of the series, when n is even. I understand that we can assume that n = 2m, where m belongs to the set of natural numbers. But the way it has been solved, that's what I don't understand. It says S(n) = S(2m), which is obvious. What I don't understand is, that result : m[2m³+4m²+5m+2], that's S(m), i.e. sum of first m terms, right? Then we put m = 2m in order to find out the sum of first 2m terms. But the book says that the expression m[2m³+4m²+5m+2] is S(2m), i.e sum of first 2m terms. How? Isn't it the sum of first m terms?
Look carefully, the sum given is $S_{2m}$, not $S_m$, so we substitute $n =2m$ in the end. Basically we are summing $2m$ terms, not $m$ terms. The expression given is for sum of $2m$ terms.