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
Proving a base case and showing the property holds for if it holds for
Finding a single case where a statement fails to disprove it. 3. Application: The Fibonacci Identity
of real numbers is defined as a if, for all indices , the following inequality holds:
∑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:
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 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
Proving a base case and showing the property holds for if it holds for stefani_problem_stefani_problem
Finding a single case where a statement fails to disprove it. 3. Application: The Fibonacci Identity fkfk+1+fk+12=fk+1(fk+fk+1)f sub k f sub k plus 1
of real numbers is defined as a if, for all indices , the following inequality holds: Contraposition: Proving "If not B, then not A
∑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:
Assuming the property is false and showing this leads to an impossibility. Contraposition: Proving "If not B, then not A."