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Question Number 190704 by mathlove last updated on 09/Apr/23

$$f(x+y)=f\left(x\right)+f\left(y\right)+x\cdot y$$$$f\left(4\right)=10findef\left(2022\right)=?$$

Answered by mahdipoor last updated on 09/Apr/23

$$f(n+1)=\left[f\left(n\right)\right]+f\left(1\right)+n=$$$$[f(n-1)+f\left(1\right)+(n-1)]+f\left(1\right)+n=...$$$$=(n+1)f\left(1\right)+\frac{n(n+1)}{2}$$$$\Rightarrow f\left(4\right)=4f\left(1\right)+\frac{3\times 4}{2}=10\Rightarrow f\left(1\right)=1$$$$f\left(2022\right)=2022f\left(1\right)+\frac{2021\times 2022}{2}=1011\times 2023$$

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