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Chapter 5: Problem 107
Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)
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Use rules of divisibility to determine whether each number given is divisible by a. 2 b. 3 c. 4 d. 5 e. 6 f. 8 g. 9 h. 10 i. 12 . 89,001
Shown in the figure is an 8-hour clock and the table for clock addition in the 8-hour clock system. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline \oplus & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline \mathbf{0} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathbf{1} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 0 \\ \hline \boldsymbol{2} & 2 & 3 & 4 & 5 & 6 & 7 & 0 & 1 \\ \hline \mathbf{3} & 3 & 4 & 5 & 6 & 7 & 0 & 1 & 2 \\ \hline \mathbf{4} & 4 & 5 & 6 & 7 & 0 & 1 & 2 & 3 \\ \hline \mathbf{5} & 5 & 6 & 7 & 0 & 1 & 2 & 3 & 4 \\ \hline \mathbf{6} & 6 & 7 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \mathbf{7} & 7 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array} $$ a. How can you tell that the set \(\\{0,1,2,3,4,5,6,7\\}\) is closed under the operation of clock addition? b. Verify the associative property: $$ (4 \oplus 6) \oplus 7=4 \oplus(6 \oplus 7) \text {. } $$ c. What is the identity element in the 8-hour clock system? d. Find the inverse of each element in the 8-hour clock system. e. Verify two cases of the commutative property: \(5 \oplus 6=6 \oplus 5\) and \(4 \oplus 7=7 \oplus 4\).
You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ 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 \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.
Find the least common multiple of the numbers. 240 and 285
Use the distributive property to simplify the radical expressions \(\sqrt{3}(5+\sqrt{3})\)
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