Solved: . The Fibonacci sequence is a sequence given by the equations:

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the fibonacci sequence is a sequence given by the equations f1 1 f2 1

Question

Complex Analysis

. The Fibonacci sequence is a sequence given by the equations:

F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn

So for example,

F3 = F2 + F1 =1+1=2

F1 = F3+ F2 = 2 +1 = 3

(a) (2 marks) Use the definition of the (n +1)t Fibonacci number to find a 2 × 2matrix A which satisfies the equation:

A\left(\begin{array}{c} F_{n} \\ F_{n+1} \end{array}\right)=\left(\begin{array}{c} F_{n+1} \\ F_{n+2} \end{array}\right)

(b) (8 marks) Compute the eigenvalues of the matrix A that you found in the previous question. For each eigenvalue, find a basis for the eigen space.

(d) 4 marks Use the equation from part (a) to prove by induction on n that for alln > 0,

A^{n}\left(\begin{array}{l} F_{1} \\ F_{2} \end{array}\right)=\left(\begin{array}{l} F_{n+1} \\ F_{n+2} \end{array}\right)

(e) (4 marks) Use parts (c) and (d) to prove following formula for the nth Fibonacci number:

F_{n}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)

Answer

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Question 39041

Complex Analysis

a) Let C be a set, and A and B be disjoint sets. Find a bijection vh: CA × CB → CAUB¸ and show that it is well-defined, injective,and surjective.
(b) Denote by |X| the cardinality of a set X.
(i) Let A, B, C, and D be sets such that |A| = |B| and |C| =|D]. Prove that
|Ac| = |Bd
(ii) Find a pair of sets A and B so that |A| < |B|, but |AN| =|B]. Find an example where B is countably infinite. (Noproof is necessary.)
(iii) Find two different explicit bijections f, g: 5N → 2N. By explicit, I mean recipes for converting elements of 5N into elements of 2N. Explain your answer graphically in terms of the trees for 2N and 5N.

Complex Analysis

2. If arctan x =y, what is cos y?

Complex Analysis

1. The tide in a river has a 60-cm variation from high tide to low tide. During a 24 hour period, there are two high tides and two low tides. The lowest tide of the river is 11cm and the tidal patterns form a sinusoidal wave. At 1 am, the tide is halfway between high and low tide and dropping.
(a) Sketch a graph of at least one period of this function. Label your graph and define your variables. Let t represent the number of hours since midnight.
(b) Write two different equations that could represent the situation above.
(c) A fisherman prefers to go fishing when the tide is approximately 25cm. Calculate one time when this occurs (give both an exact answer and an answer rounded to two decimal places).

Complex Analysis

(2) Let AABC be an isosceles triangle with AB = AC. Let BM be thealtitude from B on AC and CN be the altitude from C on AB. Show thatBM CN.

Complex Analysis

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Complex Analysis

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a. What are the dimensions of the desk, to the nearest tenth of a centimetre?
b. What is the minimum perimeter of the desk?(2 marks)

Complex Analysis

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Complex Analysis

13. An 8 m ladder is leaning against a wall. The foot of the ladder is 6 m from the wall.
a) How far up the wall does the ladder reach? Round your answer to one decimal place. Include a diagram with your solution.(3 marks)
b) What is the angle of elevation of the ladder? Round to the nearest whole number.

Complex Analysis

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Complex Analysis

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