Chapter 9

List the first five terms of the sequence \(a_{n}=\frac{15 n \cdot(-2)^{n-1}}{47}\),

List the first four terms of the sequence \(a_{n}=5.7^{n}+0.275(n-1) !\)

Give two examples of arithmetic sequences whose \(4^{\text {th }}\) terms are 9.

List the first six terms of the sequence \(a_{n}=\frac{n !}{n}\).

Give two examples of arithmetic sequences whose \(10^{\text {th }}\) terms are \(206 .\)

Consider the sequence defined by \(a_{n}=-6-8 n .\) Is \(a_{n}=-421\) a term in the sequence? Verify the result.

Find the \(5^{\text {th }}\) term of the arithmetic sequence \(\\{9 b, 5 b, b, \ldots\\}\).

What term in the sequence \(a_{n}=\frac{n^{2}+4 n+4}{2(n+2)}\) has the value \(41 ?\) Verify the result.

Find the \(11^{\text {th }}\) term of the arithmetic sequence \(\\{3 a-2 b, a+2 b,-a+6 b \ldots\\}\).

Find a recursive formula for the sequence 1,0,-1,-1,0,1,1,0 \(-1,-1,0,1,1, \ldots .\) (Hint: find a pattern for \(a_{n}\) based on the first two terms.)

Calculate the first eight terms of the sequences \(a_{n}=\frac{(n+2) !}{(n-1) !}\) and \(b_{n}=n^{3}+3 n^{2}+2 n,\) and then make a conjecture about the relationship between these two sequences.

For which terms does the finite arithmetic sequence \(\left\\{\frac{5}{2}, \frac{19}{8}, \frac{9}{4}, \ldots, \frac{1}{8}\right\\}\) have integer values?