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Find-all-functions-f-Z-R-such-that-f-n-m-nf-n-mf-m-nm-n-m-n-m-Z-




Question Number 156447 by MathSh last updated on 11/Oct/21
Find all functions  f : Z → R  such that  f(n+m)=nf(n)+mf(m)+nm-n-m  ∀n;m∈Z
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}\::\:\mathbb{Z}\:\rightarrow\:\mathbb{R}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{n}+\mathrm{m}\right)=\mathrm{nf}\left(\mathrm{n}\right)+\mathrm{mf}\left(\mathrm{m}\right)+\mathrm{nm}-\mathrm{n}-\mathrm{m} \\ $$$$\forall\mathrm{n};\mathrm{m}\in\mathbb{Z} \\ $$
Answered by ghimisi last updated on 11/Oct/21
m=1;n=0⇒f(1)=f(1)−1⇒0=−1  ????????????????
$${m}=\mathrm{1};{n}=\mathrm{0}\Rightarrow{f}\left(\mathrm{1}\right)={f}\left(\mathrm{1}\right)−\mathrm{1}\Rightarrow\mathrm{0}=−\mathrm{1} \\ $$$$???????????????? \\ $$$$ \\ $$
Commented by MathSh last updated on 11/Oct/21
Thank you dear Ser, wrong.?
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{Ser},\:\mathrm{wrong}.? \\ $$

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