recursive to explicit formula?

I think you have an algebra mistake.

The first five terms should be: a1 =1

a2 = 2-a1 = 2-1 = 1

a3=2-a2 = 2-1 = 1

and so forth

For the first sequence: I be conscious that the distinction between the numbers are 3, 5, 7, 9, 11, hence: f(a million) = 0 f(n) = f(n-a million) + 2n - a million hence, the sequence is quadratic, i.e.: f(n) = an^2 + bn + c hence: 0 = a(a million)^2 + b(a million) + c = a + b + c 3 = a(2)^2 + b(2) + c = 4a + 2b + c 8 = a(3)^2 + b(3) + c = 9a + 3b + c fixing that, i'm getting: a = a million b = 0 c = -a million hence: f(n) = n^2 - a million For the first sequence: I be conscious that the distinction between the numbers is continuously 3, hence: f(a million) = a million f(n) = f(n-a million) + 3 hence: f(n) = mn + b hence: a million = m(a million) + b 4 = m(2) + b fixing that, i'm getting: m = 3 b = -2 hence: f(n) = 3n - 2