4-4 Arithmetic Sequences
Name
Date
Determine if each sequence is arithmetic. If it is, find a20, the 20th term of the sequence.
5, 10, 20, 40, . . .
10, 7, 4, 1, ⫺2, . . .
Test for common difference, d.
10 ⫺ 5 ⫽ 5
10 ⬆ 5, so there is no
20 ⫺ 10 ⫽ 10
Test for the common difference, d.
7 ⫺ 10 ⫽ ⫺3
4 ⫺ 7 ⫽ ⫺3
so the common difference, d ⫽ ⫺3.
common difference.
an ⫽ a1 ⫹ (n ⫺ 1)d
The sequence is not arithmetic.
Write the rule
to find the nth term.
a20 ⫽ 10 ⫹ (20 ⫺ 1)(⫺3)
a20 ⫽ 10 ⫹ (19)(⫺3)
a20 ⫽ 10 ⫹ (⫺57) ⫽ ⫺47
Substitute 20 for n,
10 for a1, and ⫺3 for d.
Simplify.
Determine whether each sequence could be arithmetic.
If arithmetic, use a pattern to write the next four terms.
1. 2, 11, 20, 29, . . .
2. 1, 3, 9, 27, . . .
11 2 9; 20 11 9
29 20 9; 29 9 38
38 9 47; 47 9 56
56 9 65
arithmetic; 38, 47, 56, 65
Copyright © by William H. Sadlier, Inc. All rights reserved.
4. y ⫹ 5, 3y ⫹ 9, 5y ⫹ 13,
7y ⫹ 17, . . .
3 1 2; 9 3 6;
6苷2
not arithmetic
5. 0.112, 1.12, 11.2, 112, . . .
d 2y 4;
7y 17 (2y 4) 9y 21
9y 21 (2y 4) 11y 25
11y 25 (2y 4) 13y 29
arithmetic;
9y 21, 11y 25, 13y 29; 15y 33
1.12 0.112 1.008;
11.2 1.12 10.08
10.08 fi 1.008
not arithmetic
3. x ⫹ 2, 2x ⫹ 4, 3x ⫹ 6, 4x ⫹ 8, . . .
d x 2;
4x 8 (x 2) 5x 10
5x 10 (x 2) 6x 12
6x 12 (x 2) 7x 14
arithmetic;
5x 10, 6x 12, 7x 14; 8x 16
3 3 9
6. 4 , 2 , 4 , 3, . . .
3
12 12 3 15
d ;3 ;
4
4 4
4
4
15 3 9 ; 18 3 21
4
4 2 4
4
4
21 3 6
4
4
arithmetic; 15, 9 , 21, 6
4 2 4
Find the indicated term of each arithmetic sequence.
7. a10 of 11, 5, ⫺1, ⫺7, . . .
d 5 – 11 6
an a1 (n 1)d
a10 11 (10 1)(6)
a10 11 9(6) 11 54
a10 43
10. a50 of 0.1, 0.4, 0.7, 1.0, . . .
d 0.3; a50 0.1 (49)(0.3)
14.8
a50 14.8
8. a10 of 15, 7, ⫺1, ⫺9, . . .
d 8; a10 15 (9)(8)
57
a10 57
1 2
4
11. a9 of 3 , 3 , 1, 3 , . . .
1
1
1
d ; a9 (8)( )
3
3
3
3; a9 3
Lesson 4-4, pages 102–105.
9. a50 of 0.2, 0.4, 0.6, 0.8, . . .
d 0.2; a50 0.2 (49)(0.2)
10
a50 10
1 1 7 5
12. a10 of 8 , 2 , 8 , 4 , . . .
3
d ;
8
1
3
a10 (9)( )
8
8
7
7
; a10 2
2
Chapter 4
95
For More Practice Go To:
Write a function rule for the nth term of each arithmetic sequence.
13. 1, 4, 7, 10, . . .
d413
an 1 (n 1)3
an 1 3n 3
an 3n 2
16. ⫺7, ⫺15, ⫺23, ⫺31, . . .
d 8;
an 7 (n 1)(8)
an 7 (8n) 8
an 8n 1
9 17 25 33
19. 2 , 2 , 2 , 2 , . . .
9
(n 1)(4)
2
9
8
an 4n 2
2
1
an 4n 2
d 4; an 14. ⫺1, 1, 3, 5, . . .
15. 6, 11, 16, 21, . . .
d 2; an 1 (n 1)2
an 1 2n 2
an 2n 3
17. ⫺2.3, 1.7, 5.7, 9.7, . . .
d 4;
an 2.3 (n 1)(4)
an 2.3 4n 4
an 4n 6.3
10 19 28 37
18. ⫺7.6, ⫺9.6, ⫺11.6, ⫺13.6, . . .
d 2;
an 7.6 (n 1)(2)
an 7.6 (2n) 2
an 2n 5.6
1
1
21. 2 , 0.3, 10 , ⫺0.1, . . .
20. 3 , 3 , 3 , 3 , . . .
10
(n 1)(3)
3
9
10
3n an 3
3
1
an 3n 3
d 3; an d 5; an 6 (n 1)5
an 6 5n 5
an 5n 1
1
(n 1)(0.2)
2
1
an (–0.2n) 0.2
2
an 0.2n 0.7
d 0.2; an Solve.
23. A tractor trailer travels 315 miles the first day
of a trip. Each day thereafter it travels another
105 miles. What is the first day on which it will
have traveled more than 1000 miles?
315 (n 1)(105) . 1000; 105n 210 . 1000;
105n . 790; n . 7.5. So the 8th day is the
first day on which the tractor trailer will have
traveled more than 1000 miles.
Let w number of weeks to run 13 km;
13 2.5 (n 1)(1.5), 1.5n 1 13, n 8;
It will take Yuri 8 weeks to run 13 km.
24. Yolanda puts some money aside for an
MP3 player and saves an additional $5.50 each
month thereafter. If she saves $64.50 after one
year, how much did she originally put aside?
Make a pattern; Let x first month savings;
sequence: x, x 5.5, x 5.5(2), x 5.5(3), . . .
and a12 64.5. So a12 64.5 x 5.5(12 1);
64.5 x 60.5; x 4. So Yolanda saved
$4 on the first month.
26. 52(48)
52(50 2)
52(50) 52(2)
2600 104 2496
96
Chapter 4
25. Bhavin buys a tree sapling with a height
of 5 cm. After 6 months it has grown to
be 47 cm tall. If it continues growing at
this average rate, how tall will the tree be
14 months after Bhavin bought it?
27. 2(19)(15)
Reason logically; Find the rate of growth:
47 5 42 cm growth; 42 cm 6 mo 7 cm/mo.
a14 5 (14 1)7 5 91 96 cm.
The tree will be 96 cm tall in 14 months.
2
1
1
28. 3 3 ⫹ (6 2 ⫹ 8 3 )
(323 813) 612
2(15)(19)
(30)(19) 570
1
1
12 6 18
2
2
Copyright © by William H. Sadlier, Inc. All rights reserved.
22. Yuri is training for a marathon. He ran 2.5 km
the first week and increased his distance by
1.5 km each week. How many weeks will it
take him to run 13 km?