Having demonstrated the base case is true, then state the induction hypothesis
Multiply through by :
Now, since , can you continue?
alpha = (1+ sqrt5)/2 and beta = (1-sqrt5)/2
alpha^2 = 1 + alpha and beta^2 = 1+ beta
Use induction to prove that for all integers n >= 1 we have
alpha^n = f(n-1)+ f(n)*(alpha) and beta^n = f(n-1)+ f(n)*(beta)
I've did my base case and plug in k+1 to n but I can't get them equal to each other. Please help, I've been playing with these numbers for hours.
We may write this result as:
You see, we have arrived at which we derived from , completing the proof by induction.
Observe that this is the same as the induction hypothesis, except is replaced with . This means it is true for all .
Have you learned the analogy of climbing a ladder or falling dominoes to mathematical induction?