Tema
Q:
Exercise 3. Consider the famous Fibonacci sequence \( \left\{x_{n}\right\}_{n=1}^{\infty} \), defined by the relations
\( x_{1}=1, x_{2}=1 \), and \( x_{n}=x_{n-1}+x_{n-2} \) for \( n \geq 3 \)
(a) Compute \( x_{20} \).
\[ \quad x_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \]
(b) Use an extended Principle of Mathematical Induction in order to show that for
(c) Use the result of part (b) to compute \( x_{20} \).
Q:
Exercise 3. Consider the famous Fibonacci sequence \( \left\{x_{n}\right\}_{n=1}^{\infty} \), defined by the relations
\( x_{1}=1, x_{2}=1 \), and \( x_{n}=x_{n-1}+x_{n-2} \) for \( n \geq 3 \)
Q:
3.2) \( \frac{x-3}{\sqrt{x+3+\sqrt{12 x}}} \)
Q:
Exercise 1. For every positive integer \( n \), Show that
\[ \forall n \in \mathbb{N}^{*} \quad 1^{3}+2^{3}+\cdots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2} \]
Q:
Q Find the domain of \( f(x)=\frac{1}{\sqrt{x^{2}-16}} \)
Q:
\( y=\sqrt {\log _{\frac{1}{3}}|x|+\log _{\frac{1}{2}}|2x-1|} \)
Q:
\( y = \sqrt { \log _ { \frac { 1 } { 3 } } | x | + \log _ { \frac { 1 } { 2 } } | 2 x - 1 | } \)
Q:
\begin{tabular}{l} - determinar o termo geral de uma sucessão; \\ - classifica- una sucessão quanto à monotonia; \\ - classificar uma sucessão quanto do Limite; \\ - calcular o limite de uma sucessão; \\ - resolver problemas práticos da vida usando as \\ propriedades de Progressões Aritméticas e de \\ \hline\end{tabular}
Q:
3. Squirrel Population A grey squirrel population was intro-
duced in a certain county of Great Britain 30 years ago.
Biologists observe that the population doubles every 6 years,
and now the population is 100,000 .
(a) What was the initial size of the squirrel population?
(b) Estimate the squirrel population 10 years from now.
(c) Sketch a graph of the squirrel population.
Q:
(a) Let one parametric equation be \( x=t \). Find the parametric equation for \( y \).
\( y=t^{2}+4 t \) for \( t \) in \( (-\infty, \infty) \)
(Do not factor.)
(b) Let one parametric equation be \( x=t-2 \). Find the parametric equation for \( y \).
\( y=\square \) for \( t \) in \( (-\infty, \infty) \)
(Simplify your answer.)
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