How to find out the formula for the first n elements of a sequence of numbers?

Question

Mathematics

There is a sequence, each n-th element of which is expressed by the formula:
A(n) = b*n + c + A(n-1)
n is a natural number.
b, c are constants.
What will be the formula for the sum of the first n elements?

Answer 1

S(n) = b * (n(n+1)(2n+1)/6 + n(n+1)/2) / 2 + c * n(n+1)/2

Answer 2

A(i) = (i)*(i+1)/2*b + (i)*c
The sum of the first N coefficients before "c" is easy to calculate - an arithmetic progression,
and before "b" - there will be a sum of squares - I doubt that it can be described by some beautiful formula.

Answer 3

If we assume that A(0) = 0, then this is what I got: b * (n1 + n2 + ... nn) + n * c

Answer 4

since there are no boundary conditions, then: