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Chapter 14: Problem 5
Write the first four terms of each sequence whose general term is given. $$a_{n}=(-3)^{n}$$
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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\). [Hint: Write \(x^{2}+x+1\) as \(\left.x^{2}+(x+1) .\right]\)
Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I use binomial coefficients to expand \((a+b)^{n},\) where \(\left(\begin{array}{c}n \\ 1\end{array}\right)\) is the coefficient of the first term, \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) is the coefficient of the second term, and so on.
Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{n}{n+1} ; n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$
Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. \begin{aligned}&16, \quad\quad\quad\quad\quad 0.96(16), \quad(0.96)^{2}(16), \quad(0.96)^{3}(16), \ldots\\\&\begin{array}{|l|l|l|}\hline \begin{array}{l}\text { 1st swing } \\\\\end{array} & \begin{array}{l}\text{ 2nd swing } \\\\\end{array} & \begin{array}{l}\text { 3rd swing } \\\\\end{array} & \begin{array}{l}\text { 4th swing } \\\\\end{array} \\\\\hline\end{array}\end{aligned} After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.
Will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?
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