91Ó°ÊÓ

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=5 n-3\)

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=1-6 n\)

Show that the sum of a finite arithmetic se- \(-\) quence is 0 if and only if the last term equals the negative of the first term.

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=3(-2)^{n}\)

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=5 \cdot 3^{-n}\)

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=2^{n} n !\)

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=\frac{3^{n}}{n !}\)

Define a recursive sequence by \(a_{1}=3 \quad\) and \(\quad a_{n+1}=\frac{1}{2}\left(\frac{7}{a_{n}}+a_{n}\right)\) for \(n \geq 1 .\) Find the smallest value of \(n\) such that \(a_{n}\) agrees with \(\sqrt{7}\) for at least six digits after the decimal point.

Define a recursive sequence by \(a_{1}=6\) and \(a_{n+1}=\frac{1}{2}\left(\frac{17}{a_{n}}+a_{n}\right)\) for \(n \geq 1 .\) Find the smallest value of \(n\) such that \(a_{n}\) agrees with \(\sqrt{17}\) for at least four digits after the decimal point.