91Ó°ÊÓ

Given that \(F_{36}=14,930,352\) and \(F_{37}=24,157,817,\) (a) find \(F_{35}\). (b) find \(F_{34}\).

Fact: If we make a list of any four consecutive Fibonacci numbers, the first one times the fourth one is always equal to the third one squared minus the second one squared. (a) Verify this fact for the list \(F_{8}, F_{9}, F_{10}, F_{11}\) (b) Using the list \(F_{N}, F_{N+1}, F_{N+2}, F_{N+3},\) write this fact as a mathematical formula.

Refer to "Fibonacci-like" sequences. Fibonacci-like sequences are based on the same recursive rule as the Fibonacci sequence (from the third term on each term is the sum of the two preceding terms), but they are different in how they get started. Consider the Fibonacci-like sequence \(2,4,6,10,16,26, \ldots\) and let \(B_{N}\) denote the \(N\) th term of the sequence. (a) Find \(B_{9}\). (b) Given that \(F_{20}=6765,\) find \(B_{19}\). (c) Express \(B_{N}\) in terms of \(F_{N+1}\).

Consider the quadratic equation \(x^{2}=3 x+1\) (a) Use the quadratic formula to find the two solutions of the equation. (Remember that the equation has to be changed to standard form first.) Give the value of each solution rounded to five decimal places. (b) Find the sum of the two solutions in (a). (c) Explain why the decimal part has to be exactly the same in both solutions.

The reciprocal of \(\phi=\frac{1+\sqrt{5}}{2}\) is the irrational number \(\frac{1}{\phi}=\frac{2}{1+\sqrt{5}}\) (a) Using a calculator, compute \(\frac{1}{\phi}\) to 10 decimal places. (b) Explain why \(\frac{1}{\phi}\) has exactly the same decimal part as \(\phi\). (Hint: Show that \(\left.\frac{1}{\phi}=\phi-1 .\right)\)

The square of the golden ratio is the irrational number \(\phi^{2}=\left(\frac{1+\sqrt{5}}{2}\right)^{2}=\frac{3+\sqrt{5}}{2}\). (a) Using a calculator, compute \(\phi^{2}\) to 10 decimal places. (b) Explain why \(\phi^{2}\) has exactly the same decimal part as \(\phi\).

The Fibonacci sequence of order 3 is the sequence of numbers \(1,3,10,33,109, \ldots\) Each term in this sequence (from the third term on) equals three times the term before it plus the term two places before it; in other words, $$ A_{N}=3 A_{N-1}+A_{N-2}(N \geq 3) $$ (a) Compute \(A_{6}\). (b) Use your calculator to compute to five decimal places the ratio \(A_{6} / A_{5}\) (c) Guess the value (to five decimal places) of the ratio \(A_{N} / A_{N-1}\) when \(N>6\)

The Lucas sequence is the Fibonacci-like sequence \(1,3,4,7,11,18,29,47, \ldots\) (first introduced in Exercise 23 ). The numbers in the Lucas sequence are called the Lucas numbers, and we will use \(L_{N}\) to denote the \(N\) th Lucas number. The Lucas numbers satisfy the recursive rule \(L_{N}=L_{N-1}+L_{N-2}\) (just like the Fibonacci numbers), but start with the initial values \(L_{1}=1, L_{2}=3\). (a) Show that the Lucas numbers are related to the Fibonacci numbers by the formula \(L_{N}=2 F_{N+1}-F_{N}\). [Hint: Let \(K_{N}=2 F_{N+1}-F_{N},\) and show that the numbers \(K_{N}\) satisfy exactly the same definition as the Lucas numbers (same initial values and same recursive rule). ] (b) Show that \(\left(\frac{L_{N+1}}{L_{N}}\right) \rightarrow \phi .[\) Hint: Use (a) combined with the fact that \(\left(\frac{F_{N+1}}{F_{N}}\right) \rightarrow \phi .\) ]

Explain why the only even Fibonacci numbers are those having a subscript that is a multiple of 3 .