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 $f(n)$.

Question

anonymous · Answer

f(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
$$f \left( 3 \right)=11=3^2+2=3^2+3-1$$
put x=2,y=2
f(2+2)=f(2)+f(2)+2*2*2+1=5+5+8+1=19
$$f \left( 4 \right)=19=4^2+3=4^2+4-1$$
put x=3,y=2
f(3+2)=f(3)+f(2)+2*3*2+1=11+5+12+1=29
$$f \left( 5 \right)=29=5^2+4=5^2+5-1$$
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$$f \left( n \right)=n^2+n-1$$