The Fibonacci sequence is defined recursively as follows: $f_0=0, f_1=1, f_{f}=f_{(f−1)}+f_{(f−2)}$ for all integers $f$ with $f\ge 2$. The first few terms are $0, 1, 1, 2, 3, 5, 8, 13$. Prove that

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

for every non-negative integer $f$.

a) Give a recursive definition of the Fibonacci numbers. b) Show that

$$f_n > α^{n−2}$$

whenever n ≥ 3, where fₙ is the nth term of the Fibonacci sequence and α = (1 + √5)/2.

The Fibonacci sequence was defined by the equations f1=1, f2=1, fn=fn-1 + fn-2, n≥3. Show that each of the following statements is true. 1/fn-1 fn+1 = 1/fn-1 fn - 1/fn fn+1

A discrete-time signal can be formed from a continuous-time signal by sampling.

Write a function that represents the situation. A $900 sound system decreases in value by 9% each year.

Find the cardinal number of given of the following sets. Assume the pattern of elements continues in given part in the order given.
a.{201, 202, 203, ...... ,1100}
b.{1, 3, 5, ..... , 101}
c.{1, 2, 4, 8, 16, ..... , 1024}
d. {x|x = $k^{3}$ , k = 1, 2, 3, ..... ,100}

Prove that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue of $A^2$.

Solve for c, where c represents the length of the hypotenuse of a right triangle. Simplify the result, if possible. a.

$$c ^ { 2 } = 64$$

b.

$$c ^ { 2 } = 15$$

c.

$$c ^ { 2 } = 24$$

Analyze and define given of the following word. In this and in succeeding exercises, analysis should consist of separating the word into prefixes (if any), combining forms, and suffixes or suffix forms (if any) and giving the meaning of each. Be certain to differentiate between nouns and adjectives in your definitions. Consult a medical dictionary for the current meanings of these word. Use a separate paper if you need more room for an answer.
dipsophobia _________________

Determine the non-permissible values of x, in radians, for the given expression.

$$\frac{\sin x}{\cos x}$$

∫^5_0 f(x)dx=10 and ∫^7_5f(x)dx=3 evaluate (a) ∫^7_0𝑓(𝑥)ⅆ𝑥 (b)∫^0_5f(x)dx (c) ∫^5_5𝑓(𝑥)ⅆ𝑥 (d) ∫^5_0 3𝑓(𝑥)ⅆ𝑥

Evaluate the function for the given value of $x$.

$f(x)=3 \sqrt[5]{-x}$ for $x=32$

If you were given one of the non-included angles, $\angle C$ or $\angle B$, instead of $\angle A$, do you think everyone in your class would have constructed an identical triangle?

Find all solutions of each equation.

$$x^3+2=2 i$$