Core pure AS maths: Series

Images and question below

This is the question:


This is the solution:


The part of the solution that confuses me specifically is that f(r) is set equal to 2r + b (or in other questions, 2x + b, or 2n + b, whatever). Why 2r + b? Mathematically, where did that expression come from and/or why is it used? Sorry if it seems like a dumb question

Because its given that the sum of ??? from 1 to n is a quadratic, so ??? must be linear so
ar+b
as in the first line (note that the sum from 1 to n of 1 is n (linear) and the sum of r is n(n+1)/2 (quadratic)). Then its just choosing a and b to fit the quadratic coefficient 1 and the linear coefficient 4. On the third line the sum of ar+b from 1 to n is
a/2 n^2 + (a/2+b) n
So matching the quadratic coefficients (a/2=1) gives a=2 and linear coefficients (2/2+b=4) gives b=3. So its summing
2r+3
from 1 to n