Chapter 11

Determine whether the sum of each infinite geometric series exists. $$ -972-324-108-\dots $$

Which expression represents a series with 12 terms? F. \(\sum_{n=3}^{12} 12 n \quad\) G. \(\sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \quad\) H. \(\sum_{n=9}^{21}(3 n-6) \quad\) I. \(\sum_{n=1}^{11} \frac{n}{2}\)

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=10, a_{n}=2 a_{n-1} $$

Find the 10 th term of each geometric sequence. $$ a_{9}=-\frac{1}{3}, r=\frac{1}{2} $$

Write an explicit and a recursive formula for each sequence. \(2,4,6,8,10, \dots\)

What is the common ratio for the geometric series \(\sum_{n=1}^{10} 7\left(\frac{4}{7}\right)^{n-1} ?\) Enter your answer as a fraction.

Write the equation of each hyperbola in standard form. Sketch the graph. $$ 9 x^{2}-16 y^{2}=144 $$

Which expression represents the sum of the finite series \(13+10+7+4 ?\) I. \(\sum_{n=1}^{4}(16-3 n)\) II. \(\sum_{n=3}^{6}(22-3 n)\) III. \(\sum_{n=1}^{4}(4+3 n)\) A. I and II B. I and III C. II and III D. I, II, and III

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=-5, a_{n}=a_{n-1}-1 $$

Find the 10 th term of each geometric sequence. $$ a_{11}=-\frac{1}{3}, r=\frac{1}{2} $$

What is the common ratio in a geometric series if \(a_{2}=\frac{2}{5}\) and \(a_{5}=\frac{16}{135} ?\) Enter your answer as a fraction.

Write the equation of each hyperbola in standard form. Sketch the graph. $$ x^{2}-25 y^{2}=25 $$

The first term of an arithmetic series is \(123 .\) The common difference is 12 , and the sum 1320 . How many terms are in the series? \(\begin{array}{lllll}{\text { F. } 10} & {\text { G. } 9} & {\text { H. } 8} & {\text { J. } 7}\end{array}\)

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=-2, a_{n}=\frac{1}{2} a_{n-1} $$

Writing Describe the similarities and differences between a common difference and a common ratio.

Write an explicit and a recursive formula for each sequence. \(-5,-4,-3,-2,-1, \ldots\)

Evaluate the infinite geometric series \(\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\ldots\) Enter your answer as a fraction.

Write the equation of each hyperbola in standard form. Sketch the graph. $$ 16 x^{2}-10 y^{2}=160 $$

Write an expression for the sum of a 6 -term arithmetic sequence with first term of 3 and a common difference of 4 . Then find the sum.

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=1, a_{n}=a_{n-1}+4 $$

Find the sum of the two infinite series \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n-1}\) and \(\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n}.\)

Solve each equation. Check your solution. $$ \frac{x}{4}=\frac{x-3}{8} $$

Evaluate the series \(\sum_{n=1}^{40}\left(10-\frac{n}{2}\right)\) Show your work.

Write an explicit and a recursive formula for each sequence. \(-2,5,12,19,26,33, \dots\)

Evaluate the sum \(\sum_{n=1}^{3}\left(\frac{1}{n+1}\right)^{2} .\) Enter your answer as a decimal to the nearest hundredth.

Solve each equation. Check your solution. $$ \frac{5}{2-x}=\frac{4}{2 x+1} $$

The 30 th term of a finite arithmetic series is 4.4 . The sum of the first 30 terms is \(78 .\) What is the first term of the series?

Geometry The triangular numbers form a sequence. The diagram represents the first three triangular numbers: \(1,3,\) and \(6 .\) a. Find the fifth and sixth triangular numbers. b. Write a recursive formula for the \(n\) th triangular number. c. Is the explicit formula \(a_{n}=\frac{1}{2}\left(n^{2}+n\right)\) the correct formula for this sequence? How do you know?

Car 1 cost \(\$ 22,600\) when new and depreciated 14\(\%\) each year for 5 years. The same year, Car 2 cost \(\$ 17,500\) when new and depreciated 7\(\%\) each year for 5 years. To the nearest dollar, what was the difference in the values of the two cars after 5 years?

Solve each equation. Check your solution. $$ \frac{x}{x+1}-\frac{x}{x-3}=9 $$

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=1, r=2 $$

What is the difference between the third term in the sequence whose recursive formula is \(a_{1}=-5, a_{n}=2 a_{n-1}+1\) and the third term in the sequence whose recursive formula is \(a_{1}=-3, a_{n}=-a_{n-1}+3 ?\) $$ \begin{array}{lllll}{\text { A. } 2} & {\text { B. } 14} & {\text { C. } 20} & {\text { D. } 32}\end{array} $$

Suppose a balloon is filled with 5000 \(\mathrm{cm}^{3}\) of helium. It then loses one fourth of its helium each day. a. Write the geometric sequence that shows the amount of helium in the balloon at the start of each day for five days. b. What is the common ratio of the sequence? c. How much helium will be left in the balloon at the start of the tenth day? d. Graph the sequence. Then sketch the graph. e. Critical Thinking How does the common ratio affect the shape of the graph?

Write an explicit and a recursive formula for each sequence. \(-5,-3.5,-2,-0.5,1, \ldots\)

Evaluate each series to the given term. \(12.5+15+17.5+20+22.5+\ldots ; 7th\) term

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=1, r=5 $$

What is a recursive formula for the sequence whose explicit formula is \(a_{n}=(n+1)^{2} ?\) F. \(a_{1}=1, a_{n}=\left(a_{n-1}+1\right)^{2}\) H. \(a_{1}=n, a_{n}=a_{n-1}+n\) G. \(a_{1}=4, a_{n}=\left(\sqrt{a_{n-1}}+1\right)^{2}\) J. \(a_{1}=n^{2}, a_{n}=\left(a_{n}-1\right)^{2}+1\)

Find \(a_{1}\) for a geometric sequence with the given terms. $$ a_{5}=112 \text { and } a_{7}=448 $$

Identify the focus and directrix of each parabola. Then graph the parabola. $$ y=\frac{1}{16} x^{2} $$

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=-1, r=-1 $$

Find \(a_{1}\) for a geometric sequence with the given terms. $$ a_{9}=\frac{1}{2} \text { and } a_{12}=\frac{1}{16} $$

Write an explicit and a recursive formula for each sequence. \(1,1 \frac{1}{3}, 1 \frac{2}{3}, 2, \ldots\)

Identify the focus and directrix of each parabola. Then graph the parabola. $$ x=-\frac{1}{4} y^{2} $$

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=3, r=\frac{3}{2} $$

Which geometric sequence DOES NOT include the term 100\(?\) $$ \begin{array}{ll}{\text { A. } 5,10,20, \ldots} & {\text { B. } 337.5,225,150, \ldots} \\ {\text { C. } a_{1}=25, a_{n}=2 a_{n-1}} & {\text { D. } a_{n}=4 \cdot 5^{n}}\end{array} $$

Identify the focus and directrix of each parabola. Then graph the parabola. $$ x^{2}=-9 y $$

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=-7, r=0.1 $$

What is the product of the geometric mean of 2 and 32 and the geometric mean of 1 and 4\(?\) $$ \begin{array}{lllll}{\text { F. } 16} & {\text { G. } 19} & {\text { H. } 32} & {\text { 1. } 256}\end{array} $$

Suppose a trolley stops at a certain intersection every 14 \(\mathrm{min}\) . The first trolley of the day gets to the stop at \(6 : 43\) A.M. How long do you have to wait for a trolley if you get to the stop at \(8 : 15\) A.M.? At \(3 : 20\) P.M.?

Add or subtract. Simplify where possible. $$ \frac{7}{2 c}-\frac{2}{c^{2}} $$