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Chapter 14: Problem 33
Write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x-2 y)^{10}$$
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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. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\).
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 \(b\).
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give examples of two different arithmetic sequences whose fourth term, \(a_{4},\) is 10
Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$-1,1,-1,1, \dots$$
Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n}\). 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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