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Section 8.2 Arithmetic Sequences; Partial Sums

Before Starting this Section, Review

1 Equation of a line (Section 2.3 , page 201)
2 The general term of a sequence (Section 8.1 , page 716)
3 Partial sums (Section 8.1 , page 724)

Objectives

1 Identify an arithmetic sequence and find its common difference.
2 Find the sum of the first n terms of an arithmetic sequence.

Arithmetic Sequence

1 Identify an arithmetic sequence and find its common difference.

When the difference $(a_{n + 1} - a_{n})$ between any two consecutive terms of a sequence is always the same number, the sequence is called an arithmetic sequence. In other words, the sequence $a_{1}, a_{2}, a_{3}, a_{4}, \dots$ is an arithmetic sequence if $a_{2} - a_{1} = a_{3} - a_{2} = a_{4} - a_{3} = \dots .$

Example 1 Finding the Common Difference

Find the common difference of each arithmetic sequence.

3, 23, 43, 63, 83, …
$29, 19, 9, - 1, - 11, \dots$

Solution

The common difference is $23 - 3 = 20 (or 43 - 23, 63 - 43, or 83 - 63) .$
The common difference is $19 - 29 = - 10 (or 9 - 19, - 1 - 9, or - 11 - (- 1)) .$

Practice Problem 1

Find the common difference of the arithmetic sequence

$3, - 2, - 7, - 12, - 17, \dots .$

An arithmetic sequence can be completely specified by giving the first term $a_{1}$ and the common difference d. Let’s see why. Suppose we are told that the sequence $a_{1}, a_{2}, a_{3}, a_{4}, \dots$ is an arithmetic sequence and that $a_{1} = 5$ and $d = 4 .$ Then

$\begin{array}{l} a_{2} - a_{1} = 4, & so & a_{2} = a_{1} + 4 = 5 + 4 = 9; \\ a_{3} - a_{2} = 4, & so & a_{3} = a_{2} + 4 = 9 + 4 = 13; \\ a_{4} - a_{3} = 4, & so & a_{4} = a_{3} + 4 = 13 + 4 = 17; and so on . \end{array}$

Rewriting $d = a_{n + 1} - a_{n}$ as $a_{n + 1} = a_{n} + d,$ we have a recursive definition of an arithmetic sequence.

Consider an arithmetic sequence with first term $a_{1}$ and common difference d. Using the recursive definition $a_{n + 1} = a_{n} + d$ , for $n \geq 1$ , we write

$\begin{array}{rcll} a_{1} & = & a_{1} & The first term is a_{1} . \\ a_{2} & = & a_{1} + d & Replace n with 1 in a_{n + 1} = a_{n} + d . \\ a_{3} & = & a_{2} + d = (a_{1} + d) + d = a_{1} + 2 d & Replace a_{2} with a_{1} + d; simplify . \\ a_{4} & = & a_{3} + d = (a_{1} + 2 d) + d = a_{1} + 3 d & Replace a_{3} with a_{1} + 2 d; simplify . \\ a_{5} & = & a_{4} + d = (a_{1} + 3 d) + d = a_{1} + 4 d & Replace a_{4} with a_{1} + 3 d; simplify . \\ ⋮ \\ a_{n} & = & a_{n - 1} + d \\ = & [a_{1} + (n - 2) d] + d & Replace a_{n - 1} with a_{1} + (n - 2) d . \\ = & a_{1} + (n - 1) d & Simplify . \end{array}$

Thus, we can find the general term $a_{n}$ from the values $a_{1}$ and d.

Example 2 Finding the `n`th Term of an Arithmetic Sequence

Find an expression for the nth term of the arithmetic sequence

$3, 7, 11, 15, \dots$

Solution

Because $a_{1} = 3$ and $d = 7 - 3 = 4,$ we have

$\begin{array}{rcll} a_{n} & = & a_{1} + (n - 1) d & Formula for the n th term \\ = & 3 + (n - 1) 4 & Replace a_{1} with 3 and d with 4 . \\ = & 3 + 4 n - 4 = 4 n - 1 & Simplify . \end{array}$

Note that in the expression $a_{n} = 4 n - 1,$ we get the sequence $3, 7, 11, 15, \dots$ by substituting $n = 1, 2, 3, 4, \dots .$

Practice Problem 2

Find an expression for the nth term of the arithmetic sequence

$- 3, 1, 5, 9, 13, 17, \dots .$

We can rewrite the formula for the nth term of an arithmetic sequence as follows:

$\begin{array}{rcll} a_{n} & = & a_{1} + (n - 1) d \\ a_{n} & = & a_{1} + n d - d & Distribute . \\ a_{n} & = & d n + (a_{1} - d) & Simplify . \\ a_{n} & = & A n + B, where A = d and B = a_{1} - d . \end{array}$

We conclude that the nth term of this arithmetic sequence is the value of the linear function $f (x) = A x + B$ for $x = n$ . That is, $a_{n} = f (n)$ .

This means that the points on the graph of the arithmetic sequence lie on a line that is the graph of the linear function $f (x) = A x + B$ .

Side Note

We can use the point–slope equation of a line to get the nth term formula for an arithmetic sequence. The slope m is the common difference d, and the line passes through the point $(x_{0}, y_{0}) = (1, a_{1})$ .

$\begin{array}{rcll} y - y_{0} & = & m (x - x_{0}) \\ y - a_{1} & = & d (x - 1) \\ y & = & d (x - 1) + a_{1} \\ a_{n} & = & d (n - 1) + a_{1} \end{array}$

Example 3 Graphing an Arithmetic Sequence

Graph the arithmetic sequence

$5, 3, 1, - 1, - 3, \dots$

then use the graph to find an expression for the nth term.

Solution

To graph this sequence, we plot the points $(n, a_{n})$ for $n = 1, 2, 3, \dots$ . The first five points are

$(1, 5), (2, 3), (3, 1), (4, - 1), (5, - 3), \dots .$

We verify that the points lie on a line (see Figure 8.4). First, we find the slope of this line

$m = \frac{rise}{run} = \frac{Δ y}{Δ x} = \frac{a_{2} - a_{1}}{2 - 1} = \frac{3 - 5}{2 - 1} = - 2.$

Next, we write the point–slope equation of this line, noting that the line passes through the point $(x_{1}, y_{1}) = (1, 5)$ .

$\begin{array}{rcll} y - y_{1} & = & m (x - x_{1}) \\ y - 5 & = & (- 2) (x - 1) & Substitute x_{1} = 1, y_{1} = 5, and m = - 2 . \\ y - 5 & = & - 2 x + 2 & Simplify . \\ y & = & - 2 x + 7 & Solve for y . \end{array}$

The linear function that produces this sequence is $f (x) = - 2 x + 7,$ and the general term for this sequence is $a_{n} = f (n) = - 2 n + 7$ .

Practice Problem 3

Repeat Example 3 for the sequence $7, 4, 1, - 2, - 5, \dots .$

Example 4 Finding the Common Difference of an Arithmetic Sequence

Find the common difference d and the nth term $a_{n}$ of the arithmetic sequence whose 5th term is 15 and whose 20th term is 45.

Solution

$\begin{array}{lrcll} a_{n} & = & a_{1} + (n - 1) d & Formula for the n th term \\ 45 & = & a_{1} + (20 - 1) d & Replace n with 20; a_{n} = a_{20} = 45 . \\ (1) & 45 & = & a_{1} + 19 d & Simplify . \end{array}$

Also,

$\begin{array}{lrcll} 15 & = & a_{1} + (5 - 1) d & Replace n with 5; a_{n} = a_{5} = 15 . \\ (2) & 15 & = & a_{1} + 4 d & Simplify . \end{array}$

Side Note

Alternatively, we can compute d as the slope of the line produced by the arithmetic sequence

$\begin{array}{rcl} d = m & = & \frac{Δ y}{Δ x} \\ = & \frac{a_{20} - a_{5}}{20 - 5} \\ = & \frac{45 - 15}{20 - 5} \\ = & 2 \end{array}$

and use the point−slope equation for the point

$\begin{array}{rcl} (x_{1}, y_{1}) & = & (5, a_{5}) = (5, 15) \\ y - y_{1} & = & m (x - x_{1}) \\ y - 15 & = & 2 (x - 5) \\ y & = & 2 x + 5 \end{array}$

to get $a_{n} = 2 n + 5$ .

Solving the system of equations

${\begin{array}{lcrcl} a_{1} & + & 19 d & = & 45 & (1) \\ a_{1} & + & 4 d & = & 15 & (2) \end{array}$

gives $d = 2$ and $a_{1} = 7 .$ We substitute $a_{1} = 7$ and $d = 2$ in the formula for the nth term.

$\begin{array}{rcll} a_{n} & = & a_{1} + (n - 1) d & Formula for the n th term \\ a_{n} & = & 7 + (n - 1) 2 & Replace a_{1} with 7 and d with 2 . \\ = & 7 + 2 n - 2 = 2 n + 5 & Simplify . \end{array}$

The nth term of this sequence is given by

$a_{n} = 2 n + 5, n \geq 1 .$

Practice Problem 4

Find the common difference d and the nth term $a_{n}$ of the arithmetic sequence whose 4th term is 41 and whose 15th term is 8.

Sum of a Finite Arithmetic Sequence

2 Find the sum of the first n terms of an arithmetic sequence.

We can use the following formula in many applications involving the sum of a finite arithmetic sequence.

$1 + 2 + 3 + \dots + n = \frac{n (n + 1)}{2}$

It is said that Karl Friedrich Gauss (1777–1855) discovered this formula when he realized that any sum of numbers added in reverse order produces the same sum. Therefore, if S denotes the sum of the first n natural numbers, then

$\begin{array}{l} \underline{\begin{array}{l} S & = & 1 & + & 2 & + & 3 & + \dots + & n \\ S & = & n & + & (n - 1) & + & (n - 2) & + \dots + & 1 \end{array}} \\ \begin{array}{rcll} 2 S & = & \underset{n terms}{\underset{︸}{(n + 1) + (n + 1) + (n + 1) + \dots + (n + 1)}} & Add S + S = 2 S . \\ 2 S & = & n (n + 1) \\ S & = & \frac{n (n + 1)}{2} & Divide both sides by 2 . \end{array} \end{array}$

For example, if $n = 100,$ we have

$\begin{array}{rcl} S = 1 + 2 + 3 + \dots + 100 & = & \frac{100 (100 + 1)}{2} \\ = & \frac{100 (101)}{2} = 5050 \end{array}$

We can use this method to calculate the sum $S_{n}$ of the first n terms of any arithmetic sequence.

First, we write the sum $S_{n}$ of the first n terms:

$S_{n} = a_{1} + (a_{1} + d) + (a_{1} + 2 d) + (a_{1} + 3 d) + \dots + a_{n} .$

We can also write $S_{n}$ (with the terms in reverse order) by starting with $a_{n}$ and subtracting the common difference d:

$S_{n} = a_{n} + (a_{n} - d) + (a_{n} - 2 d) + (a_{n} - 3 d) + \dots + a_{1} .$

Adding the two equations for $S_{n}$ , we find that the d’s in the sums “drop out” (for example, $(a_{1} + d) + (a_{n} - d) = a_{1} + a_{n})$ and we have

$\begin{array}{rcl} 2 S_{n} & = & \underset{n terms}{\underset{︸}{(a_{1} + a_{n}) + (a_{1} + a_{n}) + \dots + (a_{1} + a_{n})}} \\ 2 S_{n} & = & n (a_{1} + a_{n}) \\ S_{n} & = & n (\frac{a_{1} + a_{n}}{2}) Divide both sides by 2 . \end{array}$

Example 5 Finding the Sum of Terms of a Finite Arithmetic Sequence

Consider the arithmetic sequence of numbers $1, 4, 7, \dots 25 .$ Find

The number of terms in the sequence.
The sum $1 + 4 + 7 + \dots + 25 .$

Solution

In this arithmetic sequence, $a_{1} = 1$ and $d = 3 .$ Let’s find the number of terms.

$\begin{array}{rcll} a_{n} & = & a_{1} + (n - 1) d & Formula for n th term \\ 25 & = & 1 + (n - 1) 3 & Replace a_{n} with 25, a_{1} with 1, and d with 3 . \\ 24 & = & (n - 1) 3 & Subtract 1 from both sides . \\ 8 & = & n - 1 & Divide both sides by 3 . \\ n & = & 9 & Solve for n . \end{array}$

So there are 9 terms in the sequence.
$\begin{array}{rcll} S_{n} & = & n (\frac{a_{1} + a_{n}}{2}) & Formula for S_{n} \\ S_{9} & = & 9 (\frac{1 + 25}{2}) & Replace n with 9, a_{1} with 1, and a_{n} (= a_{9}) with 25 . \\ = & 9 (13) = 117 & Simplify . \end{array}$

So $1 + 4 + 7 + \dots + 25 = 117 .$

Practice Problem 5

Find the sum $\frac{2}{3} + \frac{5}{6} + 1 + \frac{7}{6} + \frac{4}{3} + \frac{3}{2} + \frac{5}{3} + \frac{11}{6} + 2 + \frac{13}{6} .$

Example 6 Calculating the Distance Traveled by a Freely Falling Object

In the introduction to this section, we described the arithmetic sequence $16, 48, 80, 112, \dots$ whose terms gave the number of feet that freely falling space junk falls in successive seconds. For this sequence, find the following:

The common difference d.
The nth term $a_{n} .$
The distance a piece of space junk travels in 10 seconds.

Solution

The common difference $d = a_{2} - a_{1} = 48 - 16 = 32 .$
$\begin{array}{rcll} a_{n} & = & a_{1} + (n - 1) d & Formula for the n th term \\ = & 16 + (n - 1) 32 & Replace a_{1} with 16 and d with 32 . \\ = & 32 n - 16 & Simplify . \end{array}$
The sum of the first ten terms of this sequence gives the distance the piece travels in ten seconds. We find $a_{10}$ from the formula:

$\begin{array}{rcll} a_{n} & = & 32 n - 16 & From part b \\ a_{10} & = & 32 (10) - 16 & Replace n with 10 . \\ = & 304 \\ S_{n} & = & n (\frac{a_{1} + a_{n}}{2}) & Formula for S_{n} \\ S_{10} & = & 10 (\frac{16 + 304}{2}) & \begin{array}{l} Replace n with 10, a_{1} with 16, and \\ a_{n} (= a_{10}) with 304 . \end{array} \\ = & 1600 & Simplify . \end{array}$

The piece falls 1600 feet in ten seconds.

Practice Problem 6

In Example 6 , find the distance the piece travels during the 11th through 15th seconds.

Section 8.2 Exercises

Concepts and Vocabulary

If 5 is the common difference of an arithmetic sequence with general term $a_{n},$ then $a_{17} - a_{16} = \underline{}$ .
If 5 is the common difference of an arithmetic sequence with general terms $a_{n}$ and $a_{21} = 7,$ then $a_{22} = \underline{}$ .
If 14 is the term immediately following the sequence term 17 in an arithmetic sequence, then the common difference is .
If $a_{1} = 2$ and $a_{11} = 22$ for an arithmetic sequence, then the common difference, d, is .
True or False. The common difference of an arithmetic sequence is always positive.
True or False. The sum of the first n terms of an arithmetic sequence always equals n times the average of its first and nth terms.
True or False. If $a_{1}$ is the first term of an arithmetic sequence with common difference d, the nth term is given by $a_{n} = a_{1} + n d .$
True or False. The points of the graph of an arithmetic sequence with common difference d lie on a line with slope d.

Building Skills

In Exercises 9–22, determine whether each sequence is arithmetic. For those that are, find the first term $a_{1}$ and the common difference d.

1, 2, 3, 4, 5, … .
1, 3, 5, 7, 9,…
2, 5, 8, 11, 14, …
$10, 7, 4, 1, - 2, \dots$
$1, \frac{1}{2}, 0, - \frac{1}{2}, - \frac{1}{4}, \dots$
2, 4, 8, 16, 32, …
$1, - 1, 2, - 2, 3, \dots$
$- \frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \dots$
$0.6, 0.2, - 0.2, - 0.6, - 1, \dots$
2.3, 2.7, 3.1, 3.5, 3.9, …
$a_{n} = 2 n + 6$
$a_{n} = 1 - 5 n$
$a_{n} = 1 - n^{2}$
$a_{n} = 2 n^{2} - 3$

In Exercises 23–32, find an expression for the nth term of the arithmetic sequence.

5, 8, 11, 14, 17, …
4, 7, 10, 13, 16, …
$11, 6, 1, - 4, - 9, \dots$
$9, 5, 1, - 3, - 7, \dots$
$\frac{1}{2}, \frac{1}{4}, 0, - \frac{1}{4}, - \frac{1}{2}, \dots$
$\frac{2}{3}, \frac{5}{6}, 1, \frac{7}{6}, \frac{4}{3}, \dots$
$- \frac{3}{5}, - 1, - \frac{7}{5}, - \frac{9}{5}, - \frac{11}{5}, \dots$
$\frac{1}{2}, 2, \frac{7}{2}, 5, \frac{13}{2}, \dots$
$e, 3 + e, 6 + e, 9 + e, 12 + e, \dots$
$2 π, 2 (π + 2), 2 (π + 4), 2 (π + 6), 2 (π + 8) \dots$

In Exercises 33–38, graph the arithmetic sequence and use the graph to find an expression for the nth term.

$- 1, - 3, - 5, - 7, - 9, \dots$
$- 3, - 1, 1, 3, 5, \dots$
$\frac{5}{2}, 3, \frac{7}{2}, 4, \frac{9}{2}, \dots$
$\frac{5}{2}, 2, \frac{3}{2}, 1, \frac{1}{2}, \dots$
$- \frac{5}{4}, - \frac{1}{2}, \frac{1}{4}, 1, \frac{7}{4}, \dots$
$\frac{7}{4}, \frac{1}{2}, - \frac{3}{4}, - 2, - \frac{13}{4}, \dots$

In Exercises 39–44, find the common difference d and the nth term $a_{n}$ of the arithmetic sequence with the specified terms.

4th term 21; 10th term 60
3rd term 15; 21st term 87
7th term 8; 15th term $- 8$
5th term 12; 18th term $- 1$
3rd term 7; 23rd term 17
11th term $- 1;$ 31st term 5

In Exercises 45–54, find the sum of each arithmetic sequence.

$1 + 2 + 3 + \dots + 50$
$2 + 4 + 6 + \dots + 102$
$1 + 3 + 5 + \dots + 99$
$5 + 10 + 15 + \dots + 200$
$3 + 6 + 9 + \dots + 300$
$4 + 7 + 10 + \dots + 301$
$2 - 1 - 4 - \dots - 34$
$- 3 - 8 - 13 - \dots - 48$
$\frac{1}{3} + 1 + \frac{5}{3} + \dots + 7$
$\frac{3}{5} + 2 + \frac{17}{5} + \dots + \frac{101}{5}$

In Exercises 55–60, find the sum of the first n terms of the given arithmetic sequence.

$2, 7, 12, \dots; n = 50$
$8, 10, 12, \dots; n = 40$
$- 15, - 11, - 7, \dots; n = 20$
$- 20, - 13, - 6, \dots; n = 25$
$3.5, 3.7, 3.9, \dots; n = 100$
$- 7, - 6.5, - 6, \dots; n = 80$

In Exercises 61–66, find n for the given value of $a_{n}$ in each arithmetic sequence.

$a_{n} = 75; 1, 3, 5, \dots$
$a_{n} = 120; 2, 4, 6, \dots$
$a_{n} = 95; - 1, 3, 7, \dots$
$a_{n} = 83; - 5, - 3, - 1, \dots$
$a_{n} = 50 \sqrt{3}; 2 \sqrt{3}, 4 \sqrt{3}, 6 \sqrt{3}, \dots$
$a_{n} = 73 π; 3 π, 5 π, 7 π, \dots$

Applying the Concepts

In Exercises 67–75, assume that the indicated sequence is arithmetic.

Orange picking. When Eric started work as an orange picker, he picked 10 oranges in the first minute, 12 in the second minute, 14 in the third minute, and so on. How many oranges did Eric pick in the first half hour?
Exercise. Walking up a steep hill, Jan walks 60 feet in the first minute, 57 feet in the second minute, 54 feet in the third minute, and so on.
1. How far will Jan walk in the nth minute?
2. How far will Jan walk in the first 15 minutes?
Contest winner. A contest winner will receive money each month for three years. The winner receives $50 the first month, $75 the second month, $100 the third month, and so on. How much money will the winner have collected after 30 months?
Salary. Darren took a 12-month temporary job with a monthly salary that increased a fixed amount each month. He can’t recall the starting salary, but does remember that he was paid $820 at the end of the third month and $910 for his last month’s work. How much was Darren’s total pay for the entire 12 months?
Hourly wage. Antonio’s new weekend job started at an hourly wage of $12.75. He is guaranteed a raise of $25 ¢$ an hour every three months for the next four years. What will Antonio’s hourly wage be at the end of four years?
Competing job offers. Denzel is considering offers from two companies. A marketing company pays $32,500 the first year and guarantees a raise of $1300 each year; an exporting company pays $36,000 the first year, with a guaranteed raise of $400 each year. Over a five-year period, which company will pay more? How much more?
Theater seating. A theater has 25 rows of seats. The first row has 20 seats, the second row has 22 seats, the third row has 24 seats, and so on. How many seats are in the theater?
Bricks in a driveway. A brick driveway has 50 rows of bricks. The first row has 16 bricks, and the fiftieth row has 65 bricks. How many bricks does the driveway contain?
Stacked boxes. A stack of boxes has 17 rows. The bottom row has 40 boxes, and the top row has 8 boxes. How many boxes does the stack contain?
Savings. Elena wanted to teach her young daughter Sophia about saving. She gave Sophia $10 ¢$ the first day and an additional $5 ¢$ per day on each subsequent day. After a while, Sophia counted her money and found that she had saved a total of $3.25. How many days had she been saving?
Flea market. Only 3 customers showed up for the opening day of a flea market. However, 9 came the second day, and an additional 6 customers came on each subsequent day. After several days, there was a total of 192 customers. How many days had the flea market been open?
Distance. Two cars, A and B, start together in the same direction from the same place. Car A goes with a uniform speed of 60 mph. Car B goes at a speed of 50 mph in the first hour and increases its speed by 5 mph each succeeding hour. After how many hours will car B overtake car A?

In Exercises 79–82, at each stage the dots are being generated following the depicted pattern. Verify that the total number of dots at each stage is a term of an arithmetic sequence and determine the total number of dots at the 20th stage.

Beyond the Basics

In Exercises 83 and 84, show that the given sequence $a_{n}$ is an arithmetic sequence. Find the common difference d.

$a_{n} = \log (\frac{a^{n}}{b^{n - 1}})$
$a_{n} = \log ({a b}^{n})$
If the sum $S_{n}$ of the first n terms of a sequence $a_{n}$ is $3 n^{2} + 4 n,$ show that the sequence $a_{n}$ is an arithmetic sequence [Hint: $a_{n} = S_{n} - S_{n - 1};$ then show $a_{n + 1} - a_{n}$ is a constant d.]
Prove that a sequence $a_{n}$ is an arithmetic sequence if and only if the sum $S_{n}$ of its first n terms is of the form $A n^{2} + B n .$
Find the sum of all natural numbers between 45 and 100 that are divisible by 3.
Find the sum of all natural numbers between 26 and 120 that are divisible by 7.
A sequence is harmonic if the reciprocals of the terms of the sequence form an arithmetic sequence. Is the sequence $\frac{1}{2}, \frac{3}{5}, \frac{3}{4}, 1, \frac{3}{2}, 3 \dots$ harmonic? Explain your reasoning.
Consider the harmonic sequence $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \dots .$
1. Find the fourth term.
2. Find the nth term.
Show that if the numbers a, m, b are consecutive terms in an arithmetic sequence, then $m = \frac{a + b}{2} .$ We call m the arithmetic mean of a and b.
If the numbers $a, m_{1}, m_{2}, \dots, m_{k}, b$ form an arithmetic sequence, we say that the numbers $m_{1}, m_{2}, \dots, m_{k}$ are k-arithmetic means between a and b. Insert k-arithmetic means between 1 and 30 so that the sum of the resulting series is 465.

Critical Thinking / Discussion / Writing

Find the nth term of an arithmetic sequence whose first term is 10 and whose 21st term is 0.
If $a_{1}$ and d are the first term and common difference, respectively, of an arithmetic sequence, find the first term and difference of an arithmetic sequence whose terms are the negatives of the terms in the given sequence.
How many terms are in the sum $\sum_{i = 22}^{100} {17 i}^{3} ?$
The sum of the first n counting numbers is 12,403. Find the value of n.

Getting Ready for the Next Section

For each sequence in Exercises 97–100, find $\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},$ and $\frac{a_{5}}{a_{4}} .$

3, 6, 12, 24, 48
$2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \frac{32}{81}$
$4, - 12, 36, - 108, 324$
$3, - \frac{3}{2}, \frac{3}{4}, - \frac{3}{8}, \frac{3}{16}$

In Exercises 101–104, the first term $a_{1}$ and a number r are given. Write the next four terms of the sequence defined by $a_{2} = a_{1} r, a_{3} = a_{2} r, a_{4} = a_{3} r,$ and $a_{5} = a_{4} r .$

$a_{1} = 2, r = 3$
$a_{1} = 3, r = \frac{2}{5}$
$a_{1} = 1, r = - 2$
$a_{1} = 1, r = - \frac{3}{2}$

In Exercises 105–110, the nth term $a_{n}$ of a sequence is given. Find $\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},$ and $\frac{a_{5}}{a_{4}} .$

$a_{n} = 5 \cdot 2^{n}$
$a_{n} = 4 \cdot 3^{n}$
$a_{n} = 2 {(- 3)}^{n}$
$a_{n} = 5 {(- 2)}^{- n}$
Let $a_{n} = - 3 \cdot 2^{n} .$ Find $a_{n + 1}$ and $a_{n - 1} .$
Let $a_{n} = 5 {(- 3)}^{- n} .$ Find $a_{n + 1}$ and $a_{n - 1} .$

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Table of Contents for Section 8.2 Arithmetic Sequences; Partial Sums

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Table of Contents for
Section 8.2 Arithmetic Sequences; Partial Sums