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Chapter 14: Problem 14
Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(4 x-1)^{3}$$
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Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?
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? $$a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} ; n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$
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.
Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the sum of the exponents on the variables in each term.
You buy a new car for 24,000 dollars. At the end of \(n\) years, the value of your car is given by the sequence $$a_{n}=24,000\left(\frac{3}{4}\right)^{n}, \quad n=1,2,3, \dots$$ Find \(a_{5}\) and write a sentence explaining what this value represents. Describe the \(n\) th term of the sequence in terms of the value of your car at the end of each year.
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