Stefani_problem_stefani_problem
fkfk+1+fk+12=fk+1(fk+fk+1)f sub k f sub k plus 1 end-sub plus f sub k plus 1 end-sub squared equals f sub k plus 1 end-sub of open paren f sub k plus f sub k plus 1 end-sub close paren by definition: fk+1fk+2f sub k plus 1 end-sub f sub k plus 2 end-sub The identity is proven for all Resources for Further Study
Look into Monge Arrays to see how these "Gnome" properties allow for faster shortest-path algorithms in geometric graphs. stefani_problem_stefani_problem
Assuming the property is false and showing this leads to an impossibility. Contraposition: Proving "If not B, then not A." fkfk+1+fk+12=fk+1(fk+fk+1)f sub k f sub k plus 1
∑i=1k+1fi2=(∑i=1kfi2)+fk+12sum from i equals 1 to k plus 1 of f sub i squared equals open paren sum from i equals 1 to k of f sub i squared close paren plus f sub k plus 1 end-sub squared Substitute the inductive hypothesis: stefani_problem_stefani_problem