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function question

Posted February 6th, 2012 by alpha
in

Find the value of f(9).
Given:f(x+y)=f(x)+f(y)+xy and f(5)=70

‹ Integral given to SEEMOUS 2011.Need help! Mathematical induction : ›

Comments

about Cauchy function

On February 26th, 2012 Structure says:

Let $ g(x)=\frac{1}{2}x^2 $ be a solution of the functional equation and let $ f(x)=g(x)+h(x) $ then $ h(x+y)=h(x)+h(y) $ and $ h(x+x)=h(x)+h(x)=2h(x) $ Also if $ h(1)=a $; $ h(5)=5h(1) $ Then $ f(5)=h(5)+\frac{25}{2}=70 $
We get $ 5h(x)=\frac{115}{2} $ or $ h(1)=\frac{23}{2} $and so $ f(1)=12 $

$$f(x)=\frac{x(x+23)}{2}$$
$$f(9)=9*16=144$$

Other solution
$ f(x+1)=f(x)+f(1)+x $
$ f(2)=f(1)+f(1)+1 $
$ f(3)=f(2)+f(1)+2 $
$ f(4)=f(3)+f(1)+3 $
$ f(5)=f(4)+f(1)+4 $

$$f(5)=5f(1)+10$$
$$f(1)=12$$
$$f(10)=2f(5)+25$$
$$f(10)=165$$
$$f(10)=f(9)+f(1)+9$$
$$f(9)=165-12-9=144$$

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