Fibonacci and Related Sequences
For the following problem, let {Fn} be the sequence of Fibonacci num.pdf
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For the following problem, let {Fn} be the sequence of Fibonacci numbers (that is, Fn = Fn-1 +Fn-2 with seed values F1 = F2 = 1.) ie. F1=1, F2=1, F3=2, F4=3, ... hint: For the following problem, let {Fn} be the sequence of Fibonacci numbers (that is, Fn = Fn- 1 +Fn-2 with seed values F1 = F2 = 1.) ie. F1=1, F2=1, F3=2, F4=3, ? Let psi be the golden ratio and let = 1 . Then + = 1 and = -1. _____ Let psi be the golden ration and = 1 . Evaluate + Recall that . Its proof is straight forward. Dividing both sides of Fn+1 = F + Fn-1 (n > = 2) by Fn , we have F1 = 1 Take limits on both sides and let (So ). Since lim = lim , it follows that x = 1 + 1/x, or equivalently, x^2 = x + 1. Hence, we have x = psi. Solution 1/ (phi)^(16) + 1/ (psi)^(16) => [(phi)^16 + (psi)^(16)]/ (phi)^16 * (psi)^(16) => Given phi*psi = -1, substiuting the value in the denominator we get => [(phi)^16 + (psi)^(16)]/ (-1)^(16) => [(phi)^16 + (psi)^(16)] => Now for ease of convenience substitute psi = 1 - phi, to simplify the expression => [ (phi)^(16) + (1 -phi)^(16) ] phi = 1 + 1/phi substituting the values we get in the second expression => [ (phi)^(16) + 1/(phi)^16 ].
For the following problem, let {Fn} be the sequence of Fibonacci num.pdf
- For the following problem, let {Fn} be the sequence of Fibonacci numbers (that is, Fn = Fn-1 +Fn-2 with seed values F1 = F2 = 1.) ie. F1=1, F2=1, F3=2, F4=3, ... hint: For the following problem, let {Fn} be the sequence of Fibonacci numbers (that is, Fn = Fn- 1 +Fn-2 with seed values F1 = F2 = 1.) ie. F1=1, F2=1, F3=2, F4=3, ? Let psi be the golden ratio and let = 1 . Then + = 1 and = -1. _____ Let psi be the golden ration and = 1 . Evaluate + Recall that . Its proof is straight forward. Dividing both sides of Fn+1 = F + Fn-1 (n > = 2) by Fn , we have F1 = 1 Take limits on both sides and let (So ). Since lim = lim , it follows that x = 1 + 1/x, or equivalently, x^2 = x + 1. Hence, we have x = psi. Solution 1/ (phi)^(16) + 1/ (psi)^(16) => [(phi)^16 + (psi)^(16)]/ (phi)^16 * (psi)^(16) => Given phi*psi = -1, substiuting the value in the denominator we get => [(phi)^16 + (psi)^(16)]/ (-1)^(16) => [(phi)^16 + (psi)^(16)] => Now for ease of convenience substitute psi = 1 - phi, to simplify the expression => [ (phi)^(16) + (1 -phi)^(16) ] phi = 1 + 1/phi substituting the values we get in the second expression => [ (phi)^(16) + 1/(phi)^16 ]