Chapter 11

Evaluate each infinite geometric series. $$ 1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+\ldots $$

Graph each curve. Use inscribed rectangles to approximate the area under the curve for the interval and rectangle width given. $$ y=x^{2}+4,-2 \leq x \leq 2,0.5 $$

Tell whether each list is a sequence or a series. Then tell whether it is finite or infinite. $$ \frac{4}{3}, \frac{7}{3}, \frac{10}{3}, \frac{13}{3}, \frac{16}{3}, \ldots $$

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 25,50,75,100, \ldots $$

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=2 n^{2}+1 $$

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. \(1+2+3+4+\ldots ; n=1000\)

a. Graph the curve \(y=\frac{1}{3} x^{3}\) . b. Use inscribed rectangles to approximate the area under the curve for the interval \(0 \leq x \leq 3\) and rectangle width of 1 unit. c. Repeat part (b) using circumscribed rectangles. d. Find the mean of the areas you found in parts \((b)\) and (c). Of the three estimates, which best approximates the area for the interval? Explain.

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 3,-3,3,-3, \dots $$

Find the missing term of each arithmetic sequence. \(\ldots 99, \square, 66, \dots\)

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=5 n $$

Tell whether each list is a sequence or a series. Then tell whether it is finite or infinite. $$ 2.3+4.6+9.2+18.4 $$

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. \(81+27+9+3+\ldots ; n=200\)

Architecture A 20 -row theater has three sections of seating. In each section, the number of seats in a row increases by one with each successive row. The first row of the middle section has 10 seats. The first row of each of the two side sections has 4 seats. a. Find the total number of chairs in each section. Then find the total seating capacity of the theater. b. Write an arithmetic series to represent each section. c. After every five rows, the ticket price goes down by \(\$ 5 .\) Front-row tickets cost \(\$ 60 .\) What is the total amount of money generated by a full house?

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 30,35,40,45, \dots $$

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=a_{n-1}-17, \text { where } a_{1}=340 $$

Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose the first person in the chain calls four people. Then each of these people calls four others, and so on. A Make a tree diagram to show the first three stages in the telephone chain. How many calls are made at each stage? b. Write the series that represents the total number of calls made through the first six stages. c. How many employees have been notified after stage six?

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ f(x)=-x^{2}+4 $$

a. Consider the finite arithmetic series \(10+13+16+\ldots+31 .\) How many terms are in it? Explain. b. Evaluate the series.

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ -5,10,-20,40, \dots $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ y=(x-0.5)^{2}+1.75 $$

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 2,1,0.5,0.25, \ldots $$

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 5,8,11,14,17, \dots $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty}\left(\frac{1}{5}\right)^{n-1} $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ g(x)=2+3 x^{2} $$

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 5,6,8,11,15, \dots $$

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 3,6,12,24,48, \dots $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 3\left(\frac{1}{4}\right)^{n-1} $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ y=\sqrt{1+x} $$

Evaluate each series to the given term. \(2+4+6+8+\ldots ; 10\) th term

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 2,2,2,2, \dots $$

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 1,8,27,64,125, \dots $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty}\left(-\frac{1}{3}\right)^{n-1} $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ g(x)=2^{x}+1 $$

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 1,4,9,16, \dots $$

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 4,16,64,256,1024, \dots $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 7(2)^{n-1} $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ y=x^{3}+2 $$

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) \(19,683 ; \quad ; \quad ; \quad ; 243 ; \ldots\)

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 49,64,81,100,121, \ldots $$

Evaluate each series to the given term. $$ 2+3+4+5+\ldots ; 100 $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty}(-0.2)^{n-1} $$

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) 2.5, \(\quad\),\(\quad\),\(\quad\),\(202.5, \ldots\)

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ -1,1,-1,1,-1,1, \dots $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ y=-(x-1)^{2}+4 \frac{1}{3} $$

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 2(1.2)^{n-1} $$

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ h(x)=\sqrt{x^{2}} $$

Evaluate each series to the given term. $$ 0.17+0.13+0.09+0.05+\ldots ; 12 $$

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) \(12.5, \square, \square, \square, 5.12, \dots\)

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ -16,-8,-4,-2, \ldots $$

A bouncing ball reaches heights of \(16 \mathrm{cm}, 12.8 \mathrm{cm},\) and 10.24 \(\mathrm{cm}\) on three consecutive bounces. a. If the ball started at a height of \(25 \mathrm{cm},\) how many times has it bounced when it reaches a height of 16 \(\mathrm{cm} ?\) b. Write a geometric series for the downward distances the ball travels from its release at 25 \(\mathrm{cm} .\) c. Write a geometric series for the upward distances the ball travels from its first bounce. d. Find the total vertical distance the ball travels before it comes to rest.