Fermat 2002 Q22

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Problem Statement

The function $f(x)$ has the property that $f(x + y) = f(x) + f(y) + 2xy \;\;\; \forall \; x \in \mathbb{N}$. If $f(1) = 4$, then the numerical value of $f(8)$ is

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Solution

$$f(x + y) = f(x) + f(y) + 2xy \;\;\; \forall \; x \in \mathbb{N}$$ $$f(1) = 4$$ $$f(2) = f(1 + 1) = f(1) + f(1) + 2(1)(1) = 4+4+2 = 10$$ $$f(4) = f(2 + 2) = f(2) + f(2) + 2(2)(2) = 10+10+8 = 28$$ $$f(8) = f(4 + 4) = f(4) + f(4) + 2(4)(4) = 28+28+32 = 88$$