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Lesson 1.2 Exercises, pages 19–24
A
3. Use each arithmetic sequence to write the first 4 terms of an
arithmetic series.
a) 2, 4, 6, 8, . . .
b) -2, 3, 8, 13, . . .
2ⴙ4ⴙ6ⴙ8
c) 4, 0, -4, -8, . . .
ⴚ2 ⴙ 3 ⴙ 8 ⴙ 13
1 1
1
d) 2, 4, 0, - 4, Á
1
1
1
ⴙ ⴙ0ⴚ
2
4
4
4ⴙ0ⴚ4ⴚ8
4. Determine the sum of the given terms of each arithmetic series.
a) 12 + 10 + 8 + 6 + 4
b) - 2 - 4 - 6 - 8 - 10
12 ⴙ 10 ⴙ 8 ⴙ 6 ⴙ 4 ⴝ 40
ⴚ2 ⴚ 4 ⴚ 6 ⴚ 8 ⴚ 10 ⴝ ⴚ30
5. Determine the sum of the first 20 terms of each arithmetic series.
a) 3 + 7 + 11 + 15 + Á
Use: Sn ⴝ
b) -21 - 15.5 - 10 - 4.5 - Á
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute:
n ⴝ 20, t1 ⴝ 3, d ⴝ 4
S20 ⴝ
20[2(3) ⴙ 4(20 ⴚ 1)]
2
S20 ⴝ 820
Use: Sn ⴝ
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute:
n ⴝ 20, t1 ⴝ ⴚ21, d ⴝ 5.5
S20 ⴝ
20[2( ⴚ 21) ⴙ 5.5(20 ⴚ 1)]
2
S20 ⴝ 625
B
6. For each arithmetic series, determine the indicated value.
a) -4 - 11 - 18 - 25 - Á ; b) 1 + 3.5 + 6 + 8.5 + Á ;
determine S28
determine S42
Use: Sn ⴝ
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute:
n ⴝ 28, t1 ⴝ ⴚ4, d ⴝ ⴚ 7
S28 ⴝ
28[2( ⴚ 4) ⴚ 7(28 ⴚ 1)]
2
S28 ⴝ ⴚ2758
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1.2 Arithmetic Series—Solutions
Use: Sn ⴝ
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute:
n ⴝ 42, t1 ⴝ 1, d ⴝ 2.5
S42 ⴝ
42[2(1) ⴙ 2 .5(42 ⴚ 1)]
2
S42 ⴝ 2194.5
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7. Use the given data about each arithmetic series to determine the
indicated value.
a) S20 = -850 and t20 = -90;
determine t1
n(t1 ⴙ tn)
2
Use: Sn ⴝ
Substitute:
Sn ⴝ ⴚ850, n ⴝ 20, tn ⴝ ⴚ90
ⴚ850 ⴝ
20(t1 ⴚ 90)
2
ⴚ 850 ⴝ 10t1 ⴚ 900
50 ⴝ 10t1
t1 ⴝ 5
c) Sn = -126, t1 = -1, and
tn = -20; determine n
n(t1 ⴙ tn)
2
Use: Sn ⴝ
Substitute:
Sn ⴝ ⴚ126, t1 ⴝ ⴚ1, tn ⴝ ⴚ20
ⴚ126 ⴝ
n( ⴚ 1 ⴚ 20)
2
ⴚ252 ⴝ ⴚ21n
n ⴝ 12
b) S15 = 322.5 and t1 = 4;
determine d
Use: Sn ⴝ
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute:
Sn ⴝ 322.5, n ⴝ 15, t1 ⴝ 4
322.5 ⴝ
322.5
43
35
d
ⴝ
ⴝ
ⴝ
ⴝ
15[2(4) ⴙ d(15 ⴚ 1)]
2
7.5(8 ⴙ 14d)
8 ⴙ 14d
14d
2.5
d) t1 = 1.5 and t20 = 58.5;
determine S15
Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine d.
Substitute: tn ⴝ 58.5, t1 ⴝ 1.5, n ⴝ 20
58.5 ⴝ 1.5 ⴙ d(20 ⴚ 1)
57 ⴝ 19d
dⴝ3
Use: Sn ⴝ
n[2t1 ⴙ d(n ⴚ 1)]
2
Substitute: n ⴝ 15, t1 ⴝ 1.5, d ⴝ 3
15[2(1.5) ⴙ 3(15 ⴚ 1)]
2
15(3 ⴙ 42)
ⴝ
2
S15 ⴝ
S15
S15 ⴝ 337.5
8. Two hundred seventy-six students went to a powwow. The first bus
had 24 students. The numbers of students on the buses formed an
arithmetic sequence. What additional information do you need to
determine the number of buses? Explain your reasoning.
I need to know the number of students on the last bus, then I can
n(t1 ⴙ tn)
. Or, I need to know the common difference, d,
2
n[2t1 ⴙ d(n ⴚ 1)]
then I can use the rule Sn ⴝ
. In each rule, I substitute
2
use the rule Sn ⴝ
what I know, then solve for n.
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1.2 Arithmetic Series—Solutions
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9. Ryan’s grandparents loaned him the money to purchase a BMX bike.
He agreed to repay $25 at the end of the first month, $30 at the end
of the second month, $35 at the end of the third month, and so on.
Ryan repaid the loan in 12 months. How much did the bike cost?
How do you know your answer is correct?
Ryan’s repayments form an arithmetic series with 12 terms, where the
1st term is his first payment, and the common difference is $5.
n[2t1 ⴙ d(n ⴚ 1)]
2
12[2(25) ⴙ 5(12 ⴚ 1)]
ⴝ
2
Use: Sn ⴝ
S12
Substitute: n ⴝ 12, t1 ⴝ 25, d ⴝ 5
S12 ⴝ 6(50 ⴙ 55)
S12 ⴝ 630
The bike cost $630.
I used a calculator to add the 12 payments to check that the answer
is the same.
10. Determine the sum of the indicated terms of each arithmetic series.
a) 31 + 35 + 39 + Á + 107
Use tn ⴝ t1 ⴙ d(n ⴚ 1)
to determine n. Substitute:
tn ⴝ 107, t1 ⴝ 31, and d ⴝ 4
107 ⴝ 31 ⴙ 4(n ⴚ 1)
76 ⴝ 4n ⴚ 4
80 ⴝ 4n
n ⴝ 20
Use: Sn ⴝ
n(t1 ⴙ tn)
2
Substitute:
n ⴝ 20, t1 ⴝ 31, and tn ⴝ 107
S20 ⴝ
20(31 ⴙ 107)
2
S20 ⴝ 1380
b) - 13 - 10 - 7 - Á + 62
Use tn ⴝ t1 ⴙ d(n ⴚ 1)
to determine n. Substitute:
tn ⴝ 62, t1 ⴝ ⴚ13, and d ⴝ 3
62 ⴝ ⴚ13 ⴙ 3(n ⴚ 1)
75 ⴝ 3n ⴚ 3
78 ⴝ 3n
n ⴝ 26
Use: Sn ⴝ
n(t1 ⴙ tn)
2
Substitute:
n ⴝ 26, t1 ⴝ ⴚ13, and tn ⴝ 62
S26 ⴝ
26( ⴚ 13 ⴙ 62)
2
S26 ⴝ 637
11. a) Explain how this series could be arithmetic.
1 + 3 + ...
This series could be arithmetic if each term was calculated by adding 2
to the preceding term.
b) What information do you need to be certain that this is an
arithmetic series?
I need to know that the number added each time is 2.
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1.2 Arithmetic Series—Solutions
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12. An arithmetic series has S10 = 100, t1 = 1, and d = 2. How can you
use this information to determine S11 without using a rule for the
sum of an arithmetic series? What is S11?
S11 ⴝ S10 ⴙ t11
I use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine t11.
I substitute n ⴝ 11, t1 ⴝ 1, d ⴝ 2.
t11 ⴝ 1 ⴙ 2(11 ⴚ 1)
t11 ⴝ 21
Then S11 ⴝ 100 ⴙ 21
S11 ⴝ 121
13. The side lengths of a quadrilateral form an arithmetic sequence. The
perimeter is 74 cm. The longest side is 29 cm. What are the other
side lengths?
The sum of the side lengths form an arithmetic series with 4 terms,
where t4 ⴝ 29 and S4 ⴝ 74.
Use Sn ⴝ
n(t1 ⴙ tn)
to determine t1.
2
Substitute: Sn ⴝ 74, n ⴝ 4, tn ⴝ 29
74 ⴝ
4(t1 ⴙ 29)
2
74 ⴝ 2t1 ⴙ 58
16 ⴝ 2t1
t1 ⴝ 8
Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine d.
Substitute: tn ⴝ 29, t1 ⴝ 8, n ⴝ 4
29 ⴝ 8 ⴙ d(4 ⴚ 1)
21 ⴝ 3d
dⴝ7
So, the other side lengths are: 8 cm, 15 cm, and 22 cm
14. Derive a rule for the sum of the first n natural numbers:
1 + 2 + 3+ ... + n
The sum of the numbers is an arithmetic series with t1 ⴝ 1, d ⴝ 1, and tn ⴝ n.
n[2t1 ⴙ d(n ⴚ 1)]
2
n[2(1) ⴙ 1(n ⴚ 1)]
Sn ⴝ
2
n(n ⴙ 1)
Sn ⴝ
2
Use: Sn ⴝ
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Substitute: t1 ⴝ 1, d ⴝ 1
1.2 Arithmetic Series—Solutions
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15. The sum of the first 5 terms of an arithmetic series is 170. The sum
of the first 6 terms is 225. The common difference is 7. Determine
the first 4 terms of the series.
S5 ⴝ 170, S6 ⴝ 225
t6 ⴝ S6 ⴚ S5
ⴝ 225 ⴚ 170
ⴝ 55
Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine t1.
Substitute: tn ⴝ 55, d ⴝ 7, n ⴝ 6
55 ⴝ t1 ⴙ 7(6 ⴚ 1)
t1 ⴝ 20
So, t2 ⴝ 27, t3 ⴝ 34, and t4 ⴝ 41
The first 4 terms are: 20 ⴙ 27 ⴙ 34 ⴙ 41
C
16. The sum of the first n terms of an arithmetic series is: Sn = 3n2 - 8n
Determine the first 4 terms of the series.
Determine S1, S2, S3, and S4.
In Sn ⴝ 3n2 ⴚ 8n:
Substitute: n ⴝ 1
S1 ⴝ 3(1)2 ⴚ 8(1)
ⴝ ⴚ5
Substitute: n ⴝ 3
S3 ⴝ 3(3)2 ⴚ 8(3)
ⴝ3
t2 ⴝ S2 ⴚ S1
t1 ⴝ S 1
ⴝ ⴚ5
ⴝ1
Substitute: n ⴝ 2
S2 ⴝ 3(2)2 ⴚ 8(2)
ⴝ ⴚ4
Substitute: n ⴝ 4
S4 ⴝ 3(4)2 ⴚ 8(4)
ⴝ 16
t3 ⴝ S3 ⴚ S2
t4 ⴝ S4 ⴚ S3
ⴝ7
ⴝ 13
17. Each number from 1 to 60 is written on one of 60 index cards. The
cards are arranged in rows with equal lengths, and no cards are left
over. The sum of the numbers in each row is 305. How many rows
are there?
Determine the sum of the first 60 natural numbers:
1 ⴙ 2 ⴙ 3 ⴙ . . . ⴙ 60
This is an arithmetic series with t1 ⴝ 1, d ⴝ 1, and t60 ⴝ 60.
n(t1 ⴙ tn)
2
60(1 ⴙ 60)
Sn ⴝ
2
Use: Sn ⴝ
Substitute: n ⴝ 60, t1 ⴝ 1, tn ⴝ 60
Sn ⴝ 1830
The sum of the numbers in each row is 305, so the number of rows is:
1830
ⴝ6
305
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1.2 Arithmetic Series—Solutions
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18. Determine the arithmetic series that has these partial sums: S4 = 26,
S5 = 40, and S6 = 57
t6 ⴝ S6 ⴚ S5
t5 ⴝ S5 ⴚ S4
t6 ⴝ 57 ⴚ 40
t5 ⴝ 40 ⴚ 26
ⴝ 17
ⴝ 14
t4 is: t5 ⴚ 3 ⴝ 11
t3 is: t4 ⴚ 3 ⴝ 8
t1 is: t2 ⴚ 3 ⴝ 2
The arithmetic series is: 2 ⴙ 5 ⴙ 8 ⴙ 11
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d ⴝ t6 ⴚ t5
d ⴝ 17 ⴚ 14
ⴝ3
t2 is: t3 ⴚ 3 ⴝ 5
ⴙ 14 ⴙ 17 ⴙ . . .
1.2 Arithmetic Series—Solutions
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