91Ó°ÊÓ

Find an explicit formula for the \(n\) th term of the sequence satisfying \(a_{1}=0\) and \(a_{n}=2 a_{n-1}+1\) for \(n \geq 2\).

Find a formula for the general term \(a_{n}\) of each of the following sequences. \(\\{1,0,-1,0,1,0,-1,0, \ldots\\}\)

Find a formula for the general term \(a_{n}\) of each of the following sequences. \(\\{1,-1 / 3,1 / 5,-1 / 7, \ldots\\}\)

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). \(a_{1}=1\) and \(a_{n+1}=-a_{n}\) for \(n \geq 1\)

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). \(a_{1}=2\) and \(a_{n+1}=2 a_{n}\) for \(n \geq 1\)

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). \(a_{1}=1\) and \(a_{n+1}=(n+1) a_{n}\) for \(n \geq 1\)

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). \(a_{1}=2\) and \(a_{n+1}=(n+1) a_{n} / 2\) for \(n \geq 1\)

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). \(a_{1}=1\) and \(a_{n+1}=a_{n} / 2^{n}\) for \(n \geq 1\)

Plot the first \(N\) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. [T] \(a_{1}=1, \quad a_{2}=2, \quad\) and for \(\quad n \geq 2,\) \(a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right) ; \quad N=30\)

Plot the first \(N\) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges. [T] \(a_{1}=1, \quad a_{2}=2, \quad a_{3}=3\) and for \(n \geq 4,\) \(a_{n}=\frac{1}{3}\left(a_{n-1}+a_{n-2}+a_{n-3}\right), \quad N=30\)