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A prime number is an emirp ("prime" spelled backward) if it becomes a different prime number when its digits are reversed. Determine whether or not each prime number is an emirp. 43

State the distributive property of multiplication over addition and give an example.

There are two species of insects, Magicicada septendecim and Magicicada tredecim, that live in the same environment. They have a life cycle of exactly 17 and 13 years, respectively. For all but their last year, they remain in the ground feeding on the sap of tree roots. Then, in their last year, they emerge en masse from the ground as fully formed cricketlike insects, taking over the forest in a single night. They chirp loudly, mate, eat, lay eggs, then die six weeks later. (Source: Marcus du Sautoy, The Music of the Primes, HarperCollins, 2003) a. Suppose that the two species have life cycles that are not prime, say 18 and 12 years, respectively. List the set of multiples of 18 that are less than or equal to 216 . List the set of multiples of 12 that are less than or equal to 216. Over a 216-year period, how many times will the two species emerge in the same year and compete to share the forest? b. Recall that both species have evolved prime-number life cycles, 17 and 13 years, respectively. Find the least common multiple of 17 and 13 . How often will the two species have to share the forest? c. Compare your answers to parts (a) and (b). What explanation can you offer for each species having a prime number of years as the length of its life cycle?

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(7,19,31,43, \ldots\)

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(3,-6,12,-24, \ldots\)

Explain the negative exponent rule and give an example.

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by \(2019 .\)

Company A pays $$\$ 24,000$$ yearly with raises of $$\$ 1600$$ per year. Company B pays $$\$ 28,000$$ yearly with raises of $$\$ 1000$$ per year. Which company will pay more in year 10 ? How much more?

A professional baseball player signs a contract with a beginning salary of $$\$ 3,000,000$$ for the first year with an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year 2 , the athlete's salary will be \(1.04\) times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.