Methodology for Solving Recursive Functions Problems :

Question

Given that $f(x) = f(x+3)+ x^2 +x -3$ for all real numbers , and $f(1)=2$. Find $f(400)$ .

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What would be the general approach for these sorts of problems ?

Answer 1

Hint:

From the given equation,

$$f(x+3)=f(x)-(x^2+x-3)$$

and by induction,

$$f(400)=f(1)-\sum_{n=0}^{132}((3n+1)^2+(3n+1)-3)=f(1)-\sum_{n=0}^{132}(9n^2+9n-1).$$

The terms of the summation are easily found using the Faulhaber formulas.