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Chapter 14: Problem 64
What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\)
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What is a sequence? Give an example with your description.
You will develop geometric sequences that model the population growth for California and Texas, the two most-populated U.S. states. The table shows population estimates for California from 2003 through 2006 from the U.S. Census Bureau. $$\begin{array}{|c|c|c|}\hline \text { Year } & 2003 & 2004 & 2005 & 2006 \\\\\hline \text { Population in millions } & 35.48 & 35.89 & 36.13 & 36.46 \\\\\hline\end{array}$$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after 2002. c. Use your model from part (b) to project California's population, in millions, for the year 2010 . Round to two decimal places.
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January \(10,\) how many degree-days are included from January 1 to January 10?
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.
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 \(a\).
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