Question 302042
the 2nd level differences are constant (differences of differences), so this is a second order (squared) relation


a = xn^2 + yn + z


A) substituting for n=1 ___ 1 = x + y + z


B) substituting for n=2 ___ 3 = 4x + 2y + z


C) substituting for n=3 ___ 6 = 9x + 3y + z


subtracting A) from B) ___ 2 = 3x + y ___ 2 - 3x = y


subtracting A) from C) ___ 5 = 8x + 2y


substituting ___ 5 = 8x + 2(2 - 3x) ___ 5 = 8x + 4 - 6x ___ 1 = 2x ___ 1/2 = x


substituting ___ 2 - 3(1/2) = y ___ 1/2 = y


substituting ___ 1 = (1/2) + (1/2) + z ___ 0 = z


the equation is ___ a = (1/2)(n^2 + n)