Math Problems: A polynomial f(n) satisfies the equation...

Problem:

A polynomial f(n) satisfies the equation n^3 + 3n^2 + 3n + 1 = 2(f(n+1)) - f(n). Solve for f(10).

Answer: 777

Since the highest order is 3, we can assume that our function f(n) has the form
an^3 + bn^2 + cn + d

and 2(f(n+1)) has the form
2[a(n+1)^3 + b(n+1)^2 + c(n+1) + d]

After expanding, we can compare the coefficients on the left hand side with those on the right hand side:
a = 1
6a+b = 3, b = -3
6a+4b+c = 3, c = 9
2a + 2b + 2c + d = 1, d =-13

so f(n) = n^3 - 3n^2 + 9n - 13
Substitute a value of 10 for n will result in 777.