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Chapter 5: Problem 135
A franchise is owned by three people. The first owns \(\frac{5}{12}\) of the business and the second owns \(\frac{1}{4}\) of the business. What fractional part of the business is owned by the third person?
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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. \(3,-6,12,-24, \ldots\)
Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=9, r=-\frac{1}{3}\).
Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=4, r=-3\).
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.
Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-\frac{1}{8}, r=-2\)
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