91Ó°ÊÓ

In Exercises \(18-22,\) use the Binomial Theorem to find the indicated term. The term containing \(x^{\frac{7}{2}}\) in the expansion \((\sqrt{x}-3)^{8}\)

In Exercises \(22-30,\) find an explicit formula for the \(n^{\text {th }}\) term of the given sequence. Use the formulas in Equation 9.1 as needed. \(3,5,7,9, \ldots\)

In Exercises \(18-22,\) use the Binomial Theorem to find the indicated term. Use the Prinicple of Mathematical Induction to prove \(n !>2^{n}\) for \(n \geq 4\).

Find an explicit formula for the \(n^{\text {th }}\) term of the given sequence. Use the formulas in Equation 9.1 as needed. \(1, \frac{2}{3}, \frac{1}{3}, \frac{4}{27}, \ldots\)

Find an explicit formula for the \(n^{\text {th }}\) term of the given sequence. Use the formulas in Equation 9.1 as needed. \(1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \ldots\)

Find an explicit formula for the \(n^{\text {th }}\) term of the given sequence. Use the formulas in Equation 9.1 as needed. \(27,64,125,216, \ldots\)

Research the terms 'arithmetic mean' and 'geometric mean.' With the help of your classmates, show that a given term of a arithmetic sequence \(a_{k}, k \geq 2\) is the arithmetic mean of the term immediately preceding, \(a_{k-1}\) it and immediately following it, \(a_{k+1}\). State and prove an analogous result for geometric sequences.