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zmudz
 one year ago
The function \(f(n)\) is defined for all integers \(n\), such that \(f(x) + f(y) = f(x + y)  2xy  1\) for all integers \(x\) and \(y\) and \(f(1) = 1\). Find \(f(n)\).
zmudz
 one year ago
The function \(f(n)\) is defined for all integers \(n\), such that \(f(x) + f(y) = f(x + y)  2xy  1\) for all integers \(x\) and \(y\) and \(f(1) = 1\). Find \(f(n)\).

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0f(1)=1 \[f \left( x+y \right)=f \left( x \right)+f \left( y \right)+2xy+1\] put x=y=1 \[f \left( 1+1 \right)=f \left( 1 \right)+f \left( 1 \right)+2*1*1+1=1+1+2+1=5=2^2+1\] putx=2,y=1 f(2+1)=f(2)+f(1)+2*2*1+1=5+1+4+1=11 \[f \left( 3 \right)=11=3^2+2=3^2+31\] put x=2,y=2 f(2+2)=f(2)+f(2)+2*2*2+1=5+5+8+1=19 \[f \left( 4 \right)=19=4^2+3=4^2+41\] put x=3,y=2 f(3+2)=f(3)+f(2)+2*3*2+1=11+5+12+1=29 \[f \left( 5 \right)=29=5^2+4=5^2+51\] ................................................................. ..................................................................... ................................................................. \[f \left( n \right)=n^2+n1\]
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