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Chapter 14: Problem 4
Write the first four terms of each sequence whose general term is given. $$a_{n}=\left(\frac{1}{3}\right)^{n}$$
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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. There are no values of \(a\) and \(b\) such that $$(a+b)^{4}=a^{4}+b^{4}$$
Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The sequence for the number of seats per row in our movie theater as the rows move toward the back is arithmetic with \(d=1\) so people don't block the view of those in the row behind them.
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 exponents on \(a\).
What is a sequence? Give an example with your description.
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?
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