Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

11.2 Arithmetic Sequences and Series

For any arithmetic sequence, find the nth term when n is given and n when the nth term is given; and given two terms, find the common difference and construct the sequence.
Find the sum of the first n terms of an arithmetic sequence.

A sequence in which each term after the first is found by adding the same number to the preceding term is an arithmetic sequence.

Arithmetic Sequences

The sequence 2, 5, 8, 11, 14, 17, …is arithmetic because adding 3 to any term produces the next term. In other words, the difference between any term and the preceding one is 3. Arithmetic sequences are also called arithmetic progressions.

Example 1

For each of the following arithmetic sequences, identify the first term, $a_{1}$ $a_{1}$ , and the common difference, d.

4, 9, 14, 19, 24,…
34, 27, 20, 13, 6, −1, −8,…
$2, 2 \frac{1}{2}, 3, 3 \frac{1}{2}, 4, 4 \frac{1}{2}, \dots$ $2, 2 \frac{1}{2}, 3, 3 \frac{1}{2}, 4, 4 \frac{1}{2}, \dots$

Solution

The first term, $a_{1}$ $a_{1}$ , is the first term listed. To find the common difference, d, we choose any term beyond the first and subtract the preceding term from it.

SEQUENCE	FIRST TERM, $a_{1}$ $a_{1}$	COMMON DIFFERENCE, `d`
a) 4, 9, 14, 19, 24, …	4	$- 5 (9 - 4 = 5)$ $- 5 (9 - 4 = 5)$
b) 34, 27, 20, 13, 6, −1, −8,…	34	$- 7 (27 - 34 = - 7)$ $- 7 (27 - 34 = - 7)$
c) $2, 2 \frac{1}{2}, 3, 3 \frac{1}{2}, 4, 4 \frac{1}{2}, \dots$ $2, 2 \frac{1}{2}, 3, 3 \frac{1}{2}, 4, 4 \frac{1}{2}, \dots$	2	$\frac{1}{2} (2 \frac{1}{2} - 2 = \frac{1}{2})$ $\frac{1}{2} (2 \frac{1}{2} - 2 = \frac{1}{2})$

Note that we obtained the common difference by subtracting $a_{1}$ $a_{1}$ from $a_{2}$ $a_{2}$ . Had we subtracted $a_{2}$ $a_{2}$ from $a_{3}$ $a_{3}$ or $a_{3}$ $a_{3}$ from $a_{4}$ $a_{4}$ , we would have obtained the same values for d. Thus we can check by adding d to each term in a sequence to see if we progress correctly to the next term.

Check:

$4 + 5 = 9, 9 + 5 = 14, 14 + 5 = 19, 19 + 5 = 24$ $4 + 5 = 9, 9 + 5 = 14, 14 + 5 = 19, 19 + 5 = 24$
$34 + (- 7) = 27, 27 + (- 7) = 20, 20 + (- 7) = 13$ $34 + (- 7) = 27, 27 + (- 7) = 20, 20 + (- 7) = 13$ ,

$13 + (- 7) = 6, 6 + (- 7) = - 1, - 1 + (- 7) = - 8$ $13 + (- 7) = 6, 6 + (- 7) = - 1, - 1 + (- 7) = - 8$
$2 + \frac{1}{2} = 2 \frac{1}{2}, 2 \frac{1}{2} + \frac{1}{2} = 3, 3 + \frac{1}{2} = 3 \frac{1}{2}, 3 \frac{1}{2} + \frac{1}{2} = 4$ $2 + \frac{1}{2} = 2 \frac{1}{2}, 2 \frac{1}{2} + \frac{1}{2} = 3, 3 + \frac{1}{2} = 3 \frac{1}{2}, 3 \frac{1}{2} + \frac{1}{2} = 4$ ,

$4 + \frac{1}{2} = 4 \frac{1}{2}$ $4 + \frac{1}{2} = 4 \frac{1}{2}$

Now Try Exercise 1.

To find a formula for the general, or nth, term of any arithmetic sequence, we denote the common difference by d, write out the first few terms, and look for a pattern:

Generalizing, we obtain the following formula.

Example 2

Find the 14th term of the arithmetic sequence 4, 7, 10, 13, ….

Solution

We first note that $a_{1} = 4, d = 7 - 4$ $a_{1} = 4, d = 7 - 4$ , or 3, and $n = 14$ $n = 14$ . Then using the formula for the nth term, we obtain

\begin{array}{rcl} a_{n} & = & a_{1} + (n - 1) d \\ a_{14} & = & 4 + (14 - 1) \cdot 3 & Substituting \\ = & 4 + 13 \cdot 3 = 4 + 39 \\ = & 43. \end{array}

$\begin{array}{rcl} a_{n} & = & a_{1} + (n - 1) d \\ a_{14} & = & 4 + (14 - 1) \cdot 3 & Substituting \\ = & 4 + 13 \cdot 3 = 4 + 39 \\ = & 43. \end{array}$

The 14th term is 43.

Now Try Exercise 9.

Example 3

In the sequence of Example 2, which term is 301? That is, find n if $a_{n} = 301$ $a_{n} = 301$ .

Solution

We substitute 301 for $a_{n}$ $a_{n}$ , 4 for $a_{1}$ $a_{1}$ , and 3 for d in the formula for the nth term and solve for n:

\begin{array}{l} \begin{array}{lrc} a_{n} & = & a_{1} + (n - 1) d \\ 301 & = & 4 + (n - 1) \cdot 3 & Substituting \end{array} \\ \begin{array}{rcl} 301 & = & 4 + 3 n - 3 \\ 301 & = & 3 n + 1 \\ 300 & = & 3 n \\ 100 & = & n . \end{array}} Solving for n \end{array}

$\begin{array}{l} \begin{array}{lrc} a_{n} & = & a_{1} + (n - 1) d \\ 301 & = & 4 + (n - 1) \cdot 3 & Substituting \end{array} \\ \begin{array}{rcl} 301 & = & 4 + 3 n - 3 \\ 301 & = & 3 n + 1 \\ 300 & = & 3 n \\ 100 & = & n . \end{array}} Solving for n \end{array}$

The term 301 is the 100th term of the sequence.

Now Try Exercise 15.

Given two terms and their places in an arithmetic sequence, we can construct the sequence.

Example 4

The 3rd term of an arithmetic sequence is 8, and the 16th term is 47. Find $a_{1}$ $a_{1}$ and d and construct the sequence.

Solution

We know that $a_{3} = 8$ $a_{3} = 8$ and $a_{16} = 47$ $a_{16} = 47$ . Thus we would have to add d 13 times to get from 8 to 47. That is,

\begin{array}{l} 8 + 13 d = 47. & a_{3} and a_{16} are 16 - 3, or 13, terms apart . \end{array}

$\begin{array}{l} 8 + 13 d = 47. & a_{3} and a_{16} are 16 - 3, or 13, terms apart . \end{array}$

Solving $8 + 13 d = 47$ $8 + 13 d = 47$ , we obtain

\begin{array}{rcl} 13 d & = & 39 \\ d & = & 3. \end{array}

$\begin{array}{rcl} 13 d & = & 39 \\ d & = & 3. \end{array}$

Since $a_{3} = 8$ $a_{3} = 8$ , we subtract d twice to get $a_{1}$ $a_{1}$ . Thus,

\begin{array}{rcl} a_{1} = 8 - 2 \cdot 3 = 2. & a_{1} and a_{3} are 3 - 1, or 2, terms apart . \end{array}

$\begin{array}{rcl} a_{1} = 8 - 2 \cdot 3 = 2. & a_{1} and a_{3} are 3 - 1, or 2, terms apart . \end{array}$

The sequence is 2, 5, 8, 11, …. Note that we could also subtract d 15 times from $a_{16}$ $a_{16}$ in order to find $a_{1}$ $a_{1}$ .

Now Try Exercise 23.

In general, d should be subtracted $n - 1$ $n - 1$ times from $a_{n}$ $a_{n}$ in order to find $a_{1}$ $a_{1}$ .

Sum of the First `n` Terms of an Arithmetic Sequence

Consider the arithmetic sequence

3, 5, 7, 9, \dots .

$3, 5, 7, 9, \dots .$

When we add the first 4 terms of the sequence, we get $S_{4}$ $S_{4}$ , which is

3 + 5 + 7 + 9, or 24.

$3 + 5 + 7 + 9, or 24.$

This sum is called an arithmetic series. To find a formula for the sum of the first n terms, $S_{n}$ $S_{n}$ , of an arithmetic sequence, we first denote an arithmetic sequence, as follows:

\begin{array}{l} \begin{array}{l} This term is two terms back from the \\ last . If you add d to this term, the result \\ is the next-to-last term, a_{n} - d . \end{array} \\ a_{1}, (a_{1} + d), (a_{1} + 2 d), \dots, \overset{↓}{\overset{︷}{(a_{n} - 2 d),}} \underset{↑}{\underset{︸}{(a_{n} - d),}} a_{n} . \\ \begin{array}{l} This is the next-to-last term . If you \\ add d to this term, the result is a_{n} . \end{array} \end{array}

$\begin{array}{l} \begin{array}{l} This term is two terms back from the \\ last . If you add d to this term, the result \\ is the next-to-last term, a_{n} - d . \end{array} \\ a_{1}, (a_{1} + d), (a_{1} + 2 d), \dots, \overset{↓}{\overset{︷}{(a_{n} - 2 d),}} \underset{↑}{\underset{︸}{(a_{n} - d),}} a_{n} . \\ \begin{array}{l} This is the next-to-last term . If you \\ add d to this term, the result is a_{n} . \end{array} \end{array}$

Then $S_{n}$ $S_{n}$ is given by

\begin{array}{l} S_{n} & = & a_{1} + (a_{1} + d) + (a_{1} + 2 d) + \dots + (a_{n} - 2 d) \\ + (a_{n} - d) + a_{n} . \end{array}

$\begin{array}{l} S_{n} & = & a_{1} + (a_{1} + d) + (a_{1} + 2 d) + \dots + (a_{n} - 2 d) \\ + (a_{n} - d) + a_{n} . \end{array}$ (1)

Reversing the order of the addition gives us

\begin{array}{rcl} S_{n} & = & a_{n} + (a_{n} - d) + (a_{n} - 2 d) + \dots + (a_{1} + 2 d) \\ + (a_{1} + d) + a_{1} . \end{array}

$\begin{array}{rcl} S_{n} & = & a_{n} + (a_{n} - d) + (a_{n} - 2 d) + \dots + (a_{1} + 2 d) \\ + (a_{1} + d) + a_{1} . \end{array}$ (2)

If we add corresponding terms of each side of equations (1) and (2), we get

\begin{array}{rcl} 2 S_{n} & = & [a_{1} + a_{n}] + [(a_{1} + d) + (a_{n} - d)] + [(a_{1} + 2 d) + (a_{n} - 2 d)] \\ + \dots + [(a_{n} - 2 d) + (a_{1} + 2 d)] \\ + [(a_{n} - d) + (a_{1} + d)] + [a_{n} + a_{1}] . \end{array}

$\begin{array}{rcl} 2 S_{n} & = & [a_{1} + a_{n}] + [(a_{1} + d) + (a_{n} - d)] + [(a_{1} + 2 d) + (a_{n} - 2 d)] \\ + \dots + [(a_{n} - 2 d) + (a_{1} + 2 d)] \\ + [(a_{n} - d) + (a_{1} + d)] + [a_{n} + a_{1}] . \end{array}$

In the expression for $2 S_{n}$ $2 S_{n}$ , there are n expressions in square brackets. Each of these expressions is equivalent to $a_{1} + a_{n}$ $a_{1} + a_{n}$ . Thus the expression for $2 S_{n}$ $2 S_{n}$ can be written in simplified form as

\begin{array}{l} 2 S_{n} & = & [a_{1} + a_{n}] + [a_{1} + a_{n}] + [a_{1} + a_{n}] + \dots + [a_{n} + a_{1}] \\ + [a_{n} + a_{1}] + [a_{n} + a_{1}] . \end{array}

$\begin{array}{l} 2 S_{n} & = & [a_{1} + a_{n}] + [a_{1} + a_{n}] + [a_{1} + a_{n}] + \dots + [a_{n} + a_{1}] \\ + [a_{n} + a_{1}] + [a_{n} + a_{1}] . \end{array}$

Since $a_{1} + a_{n}$ $a_{1} + a_{n}$ is being added n times, it follows that

2 S_{n} = n (a_{1} + a_{n}),

$2 S_{n} = n (a_{1} + a_{n}),$

from which we get the following formula.

Example 5

Find the sum of the first 100 natural numbers.

Solution

The sum is

1 + 2 + 3 + \dots + 99 + 100.

$1 + 2 + 3 + \dots + 99 + 100.$

This is the sum of the first 100 terms of the arithmetic sequence for which

\begin{array}{l} a_{1} = 1 & a_{n} = 100, & and & n = 100. \end{array}

$\begin{array}{l} a_{1} = 1 & a_{n} = 100, & and & n = 100. \end{array}$

Thus substituting into the formula

S_{n} = \frac{n}{2} (a_{1} + a_{n}),

$S_{n} = \frac{n}{2} (a_{1} + a_{n}),$

we get

S_{100} = \frac{100}{2} (1 + 100) = 50 (101) = 5050.

$S_{100} = \frac{100}{2} (1 + 100) = 50 (101) = 5050.$

The sum of the first 100 natural numbers is 5050.

Now Try Exercise 27.

Example 6

Find the sum of the first 15 terms of the arithmetic sequence 4, 7, 10, 13, ….

Solution

Note that $a_{1} = 4, d = 3$ $a_{1} = 4, d = 3$ , and $n = 15$ $n = 15$ . Before using the formula

S_{n} = \frac{n}{2} (a_{1} + a_{n}),

$S_{n} = \frac{n}{2} (a_{1} + a_{n}),$

we find the last term, $a_{15}$ $a_{15}$ :

\begin{array}{rcl} a_{15} & = & 4 + (15 - 1) 3 & Substituting into the formula a_{n} = a_{1} + (n - 1) d \\ = & 4 + 14 \cdot 3 = 46. \end{array}

$\begin{array}{rcl} a_{15} & = & 4 + (15 - 1) 3 & Substituting into the formula a_{n} = a_{1} + (n - 1) d \\ = & 4 + 14 \cdot 3 = 46. \end{array}$

Thus,

S_{15} = \frac{15}{2} (4 + 46) = \frac{15}{2} (50) = 375.

$S_{15} = \frac{15}{2} (4 + 46) = \frac{15}{2} (50) = 375.$

The sum of the first 15 terms is 375.

Now Try Exercise 25.

Example 7

Find the sum: $\sum_{k = 1}^{130} (4 k + 5)$ $\sum_{k = 1}^{130} (4 k + 5)$ .

Solution

It is helpful to first write out a few terms:

9 + 13 + 17 + \dots .

$9 + 13 + 17 + \dots .$

It appears that this is an arithmetic series coming from an arithmetic sequence with $a_{1} = 9, d = 4$ $a_{1} = 9, d = 4$ , and $n = 130$ $n = 130$ . Before using the formula

S_{n} = \frac{n}{2} (a_{1} + a_{n}),

$S_{n} = \frac{n}{2} (a_{1} + a_{n}),$

we find the last term, $a_{130}$ $a_{130}$ :

\begin{array}{rcl} a_{130} & = & 4 \cdot 130 + 5 & The k th term is 4 k + 5. \\ = & 520 + 5 \\ = & 525. \end{array}

$\begin{array}{rcl} a_{130} & = & 4 \cdot 130 + 5 & The k th term is 4 k + 5. \\ = & 520 + 5 \\ = & 525. \end{array}$

Thus,

\begin{array}{rcl} S_{130} & = & \frac{130}{2} (9 + 525) & Substituting into S_{n} = \frac{n}{2} (a_{1} + a_{n}) \\ = & 34, 710. \end{array}

$\begin{array}{rcl} S_{130} & = & \frac{130}{2} (9 + 525) & Substituting into S_{n} = \frac{n}{2} (a_{1} + a_{n}) \\ = & 34, 710. \end{array}$

Now Try Exercise 33.

Applications

The translation of some applications and problem-solving situations may involve arithmetic sequences or series. We consider some examples.

Example 8

Hourly Wages. Kendall accepts a job, starting with an hourly wage of $14.25, and is promised a raise of 15¢ per hour every 2 months for 5 years. At the end of 5 years, what will Kendall’s hourly wage be?

Solution

It helps to first write down the hourly wage for several 2-month time periods:

Beginning:	$14.25,
After 2 months:	$14.40,
After 4 months:	$14.55,
and so on.

What appears is a sequence of numbers: 14.25, 14.40, 14.55, …. This sequence is arithmetic, because adding 0.15 each time gives us the next term.

We want to find the last term of an arithmetic sequence, so we use the formula $a_{n} = a_{1} + (n - 1) d$ $a_{n} = a_{1} + (n - 1) d$ . We know that $a_{1} = 14.25$ $a_{1} = 14.25$ and $d = 0.15$ $d = 0.15$ , but what is n? That is, how many terms are in the sequence? Each year there are $12 / 2$ $12 / 2$ , or 6 raises, since Kendall gets a raise every 2 months. There are 5 years, so the total number of raises will be $5 \cdot 6$ $5 \cdot 6$ , or 30. Thus there will be 31 terms: the original wage and 30 increased rates.

Substituting in the formula $a_{n} = a_{1} + (n - 1) d$ $a_{n} = a_{1} + (n - 1) d$ gives us

\begin{array}{l} a_{31} & = & 14.25 + (31 - 1) \cdot 0.15 \\ = & 18.75. \end{array}

$\begin{array}{l} a_{31} & = & 14.25 + (31 - 1) \cdot 0.15 \\ = & 18.75. \end{array}$

Thus, at the end of 5 years, Kendall’s hourly wage will be $18.75.

Now Try Exercise 43.

The calculations in Example 8 could be done in a number of ways. There is often a variety of ways in which a problem can be solved. In this chapter, we concentrate on the use of sequences and series and their related formulas.

Example 9

Total in a Stack. A stack of electric poles has 30 poles in the bottom row. There are 29 poles in the second row, 28 in the next row, and so on. How many poles are in the stack if there are 5 poles in the top row?

Solution

A drawing will help in this case. The following figure shows the ends of the poles and the way in which they stack.

Since the number of poles decreases from 30 in a row up to 5 in the top row, there must be 26 rows. We want the sum

30 + 29 + 28 + \dots + 5.

$30 + 29 + 28 + \dots + 5.$

Thus we have an arithmetic series. We use the formula

S_{n} = \frac{n}{2} (a_{1} + a_{n}),

$S_{n} = \frac{n}{2} (a_{1} + a_{n}),$

with $n = 26, a_{1} = 30$ $n = 26, a_{1} = 30$ , and $a_{26} = 5$ $a_{26} = 5$ .

Substituting, we get

S_{26} = \frac{26}{2} (30 + 5) = 455.

$S_{26} = \frac{26}{2} (30 + 5) = 455.$

There are 455 poles in the stack.

Now Try Exercise 39.

11.2 Exercise Set

Find the first term and the common difference.

1. 3, 8, 13, 18, …
2. $1.08, $1.16, $1.24, $1.32,…
3. 9, 5, 1, −3,…
4. −8, −5, −2, 1, 4,…
5. $\frac{3}{2}, \frac{9}{4}, 3, \frac{15}{4}, \dots$ $\frac{3}{2}, \frac{9}{4}, 3, \frac{15}{4}, \dots$
6. $\frac{3}{5}, \frac{1}{10}, - \frac{2}{5}, \dots$ $\frac{3}{5}, \frac{1}{10}, - \frac{2}{5}, \dots$
7. $316, $313, $310, $307, …
11. Find the 11th term of the arithmetic sequence 0.07, 0.12, 0.17, ….
9. Find the 12th term of the arithmetic sequence 2, 6, 10, ….
10. Find the 17th term of the arithmetic sequence 7, 4, 1, ….
11. Find the 14th term of the arithmetic sequence $3, \frac{7}{3}, \frac{5}{3}, \dots$ $3, \frac{7}{3}, \frac{5}{3}, \dots$ .
12. Find the 13th term of the arithmetic sequence $1200, $964.32, $728.64, ….
13. Find the 10th term of the arithmetic sequence $2345.78, $2967.54, $3589.30, ….
14. In the sequence of Exercise 8, what term is the number 1.67?
15. In the sequence of Exercise 9, what term is the number 106?
16. In the sequence of Exercise 10, what term is $- 296$ $- 296$ ?
17. In the sequence of Exercise 11, what term is $- 27$ $- 27$ ?
18. Find $a_{20}$ $a_{20}$ when $a_{1} = 14$ $a_{1} = 14$ and $d = - 3$ $d = - 3$ .
19. Find $a_{1}$ $a_{1}$ when $d = 4$ $d = 4$ and $a_{8} = 33$ $a_{8} = 33$ .
20. Find d when $a_{1} = 8$ $a_{1} = 8$ and $a_{11} = 26$ $a_{11} = 26$ .
21. Find n when $a_{1} = 25, d = - 14$ $a_{1} = 25, d = - 14$ , and $a_{n} = - 507$ $a_{n} = - 507$ .
22. In an arithmetic sequence, $a_{17} = - 40$ $a_{17} = - 40$ and $a_{28} = - 73$ $a_{28} = - 73$ . Find $a_{1}$ $a_{1}$ and d. Write the first 5 terms of the sequence.
23. In an arithmetic sequence, $a_{17} = \frac{25}{3}$ $a_{17} = \frac{25}{3}$ and $a_{32} = \frac{95}{6}$ $a_{32} = \frac{95}{6}$ . Find $a_{1}$ $a_{1}$ and d. Write the first 5 terms of the sequence.
24. Find the sum of the first 14 terms of the series $11 + 7 + 3 + \dots$ $11 + 7 + 3 + \dots$ .
25. Find the sum of the first 20 terms of the series $5 + 8 + 11 + 14 + \dots$ $5 + 8 + 11 + 14 + \dots$ .
26. Find the sum of the first 300 natural numbers.
27. Find the sum of the first 400 even natural numbers.
28. Find the sum of the odd numbers 1 to 199, inclusive.
29. Find the sum of the multiples of 7 from 7 to 98, inclusive.
30. Find the sum of all multiples of 4 that are between 14 and 523.
31. If an arithmetic series has $a_{1} = 2, d = 5$ $a_{1} = 2, d = 5$ , and $n = 20$ $n = 20$ , what is $S_{n}$ $S_{n}$ ?
32. If an arithmetic series has $a_{1} = 7, d = - 3$ $a_{1} = 7, d = - 3$ , and $n = 32$ $n = 32$ , what is $S_{n}$ $S_{n}$ ?

Find the sum.

33. $\sum_{k = 1}^{40} (2 k + 3)$ $\sum_{k = 1}^{40} (2 k + 3)$
34. $\sum_{k = 5}^{20} 8 k$ $\sum_{k = 5}^{20} 8 k$
35. $\sum_{k = 0}^{19} \frac{k - 3}{4}$ $\sum_{k = 0}^{19} \frac{k - 3}{4}$
36. $\sum_{k = 2}^{50} (2000 - 3 k)$ $\sum_{k = 2}^{50} (2000 - 3 k)$
37. $\sum_{k = 12}^{57} \frac{7 - 4 k}{13}$ $\sum_{k = 12}^{57} \frac{7 - 4 k}{13}$
38. $\sum_{k = 101}^{200} (1.14 k - 2.8) - \sum_{k = 1}^{5} (\frac{k + 4}{10})$ $\sum_{k = 101}^{200} (1.14 k - 2.8) - \sum_{k = 1}^{5} (\frac{k + 4}{10})$
39. Total Savings. If 10¢ is saved on October 1, 20¢ is saved on October 2, 30¢ on October 3, and so on, how much is saved altogether during the 31 days of October?
40. Stacking Poles. How many poles will be in a stack of telephone poles if there are 50 in the first layer, 49 in the second, and so on, with 6 in the top layer?
41. Auditorium Seating. Auditoriums are often built with more seats per row as the rows move toward the back. Suppose that the first balcony of a theater has 28 seats in the first row, 32 in the second, 36 in the third, and so on, for 20 rows. How many seats are in the first balcony altogether?
42. Investment Return. Brett sets up an investment situation for a client that will return $5000 the first year, $6125 the second year, $7250 the third year, and so on, for 25 years. How much is received from the investment altogether?
43. Parachutist Free Fall. When a parachutist jumps from an airplane, the distances, in feet, that the parachutist falls in each successive second before pulling the ripcord to release the parachute are as follows:

$16, 48, 80, 112, 144, \dots .$ $16, 48, 80, 112, 144, \dots .$

Is this sequence arithmetic? What is the common difference? What is the total distance fallen in 10 sec?

44. Lightning Distance. The following table lists the distance, in miles, from lightning $d_{n}$ $d_{n}$ when thunder is heard n seconds after lightning is seen. Is this sequence arithmetic? What is the common difference?

`n` (in seconds)	$d_{n}$ $d_{n}$ (in miles)
5	1
6	1.2
7	1.4
8	1.6
9	1.8
10	2

45. Garden Plantings. A gardener is making a planting in the shape of a trapezoid. It will have 35 plants in the front row, 31 in the second row, 27 in the third row, and so on. If the pattern is consistent, how many plants will there be in the last row? How many plants are there altogether?
46. Band Formation. A formation of a marching band has 10 marchers in the front row, 12 in the second row, 14 in the third row, and so on, for 8 rows. How many marchers are in the last row? How many marchers are there altogether?
47. Raw Material Production. In a manufacturing process, it took 3 units of raw materials to produce 1 unit of a product. The raw material needs thus formed the sequence

$3, 6, 9, \dots, 3 n, \dots .$ $3, 6, 9, \dots, 3 n, \dots .$

Is this sequence arithmetic? What is the common difference?

Skill Maintenance

Solve. [6.1], [6.3], [6.5], [6.6]

48. $\begin{array}{rcrcl} 7 x & - & 2 y & = & 4, \\ x & + & 3 y & = & 17 \end{array}$ $\begin{array}{rcrcl} 7 x & - & 2 y & = & 4, \\ x & + & 3 y & = & 17 \end{array}$
49. $\begin{array}{rcrcrcl} 2 x & + & y & + & 3 z & = & 12, \\ x & - & 3 y & + & 2 z & = & 11, \\ x & + & 2 y & - & 4 z & = & - 4 \end{array}$ $\begin{array}{rcrcrcl} 2 x & + & y & + & 3 z & = & 12, \\ x & - & 3 y & + & 2 z & = & 11, \\ x & + & 2 y & - & 4 z & = & - 4 \end{array}$
50. Find the vertices and the foci of the ellipse with the equation $9 x^{2} + 16 y^{2} = 144$ $9 x^{2} + 16 y^{2} = 144$ . [10.2]
51. Find an equation of the ellipse with vertices (0, −5) and (0, 5) and minor axis of length 4. [10.2]

Synthesis

52. Straight-Line Depreciation. A company buys an office machine for $5200 on January 1 of a given year. The machine is expected to last for 8 years, at the end of which time its trade-in value, or salvage value, will be $1100. If the company’s accountant figures the decline in value to be the same each year, then its book values, or salvage values, after t years, $0 \leq t \leq 8$ $0 \leq t \leq 8$ , form an arithmetic sequence given by

$a_{t} = C - t (\frac{C - S}{N}),$ $a_{t} = C - t (\frac{C - S}{N}),$

where C is the original cost of the item ($5200), N is the number of years of expected life (8), and S is the salvage value ($1100).
1. Find the formula for $a_{t}$ $a_{t}$ for the straight-line depreciation of the office machine.
2. Find the salvage value after 0 year, 1 year, 2 years, 3 years, 4 years, 7 years, and 8 years.
53. Find a formula for the sum of the first n odd natural numbers:

$1 + 3 + 5 + \dots + (2 n - 1) .$ $1 + 3 + 5 + \dots + (2 n - 1) .$
54. Find three numbers in an arithmetic sequence such that the sum of the first and the third is 10 and the product of the first and the second is 15.
55. Find the first term and the common difference for the arithmetic sequence for which

$a_{2} = 40 - 3 q and a_{4} = 10 p + q .$ $a_{2} = 40 - 3 q and a_{4} = 10 p + q .$

If p, m, and q form an arithmetic sequence, it can be shown that $m = (p + q) / 2$ $m = (p + q) / 2$ . The number m is the arithmetic mean, or average, of p and q. Given two numbers p and q, if we find k other numbers $m_{1}, m_{2}, \dots, m_{k}$ $m_{1}, m_{2}, \dots, m_{k}$ such that

p, m_{1}, m_{2}, \dots, m_{k}, q

$p, m_{1}, m_{2}, \dots, m_{k}, q$

forms an arithmetic sequence, we say that we have “inserted k arithmetic means between p and q.”

56. Insert three arithmetic means between −3 and 5.
57. Insert four arithmetic means between 4 and 13.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for 11.2 Arithmetic Sequences and Series

Create new playlist

Sign In

Sign Up

Table of Contents for
11.2 Arithmetic Sequences and Series