Chapter 8

Write the first five terms of the sequence whose first term is 9 and whose general term is a. \(-\left\\{\begin{array}{ll}\frac{a_{-1}-1}{2} & \text { if } a_{-1} \text { is even } \\ 3 a_{-1}+5 & \text { i is } a_{-1} \text { is odd }\end{array}\right.\) for \(n \geq 2\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association, which publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the Internet or the research department of your library, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}\) \(a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4}\). What do you observe?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I modeled California's population growth with a geometric sequence, so my model is an exponential function whose domain is the set of natural numbers.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

Use the formula \(a_{n}-4+(n-1)(-7)\) to find the eighth term of the sequence \(4,-3,-10, \ldots\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}.\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 10-5+\frac{5}{2}-\frac{5}{4}+\dots-\frac{10}{1-\frac{1}{2}} $$

You are now 25 years old and would like to retire at age 55 with a retirement fund of \(\$ 1,000,000 .\) How much should you deposit at the end of each month for the next 30 years in an IRA paying \(10 \%\) annual interest compounded monthly to achieve your goal? Round to the nearest dollar.

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-3: \text { Show that } 1+2+3-\frac{3(3+1)}{2}.$$

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-5 \text { : Show that } 1+2+3+4+5-\frac{5(5+1)}{2}.$$

Exercises will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{k(k+1)(2 k+1)}{6}+(k+1)^{2}.$$