Name
Class
Date
Reteaching
9.2/9.4
Arithmetic Sequences
The explicit formula for the nth term of an arithmetic sequence is
an = a + (n − 1)d.
• a is the starting value and d is the common difference.
• n is always greater than or equal to 1.
• You can write the sequence as a, a + d, a + 2d, a + 3d, . . .
Find the 15th term of an arithmetic sequence whose first three terms are
20, 16.5, and 13.
20 − 16.5 = 3.5
16.5 − 13 = 3.5
an = a + (n − 1) d
a15 = 20 + (15 − 1)(−3.5)
First, find the common difference. The difference between
consecutive terms is 3.5. The sequence decreases. The common
difference is −3.5.
Use the explicit formula.
Substitute a = 20, n = 15, and d = −3.5.
= 20 + (14)(−3.5)
Subtract within parentheses.
= 20 + −49
Multiply.
= −29
The 15th term is −29.
Check the answer. Write a1, a2, . . , a15 down the left side of your paper. Start with a1 =
20. Subtract 3.5 and record 16.5 next to a2. Continue until you find a15.
Exercises
Find the 25th term of each sequence.
1. 20, 18, 16, 14, . . .
2. 0.0057, 0.0060, 0.0063, . . .
3. 4, 0, −4, −8, . . .
4. 0.2, 0.7, 1.2, 1.7, . . .
5. −10, −8.8, −7.6, −6.4, . . .
6. 22, 26, 30, 34, . . .
To solve word problems that involve arithmetic sequences, identify the common
difference d, the starting value a, and the number of terms in the sequence n.
As a part-time home health care aide, you are paid a weekly salary plus a fixed
fuel fee for every patient you visit. You receive $240 in a week that you visit 1
patient. You receive $250 in a week that you visit 2 patients. How much will you
receive if you visit 12 patients in 1 week?
d = a2 − a1 = 250 − 240 = 10
The common difference is the difference between two consecutive
terms. You receive $10 per visit.
a = 240
Identify the starting value. You receive $240 for a week with 1 visit.
n = 12
You want to find the earnings in a week in which you visit 12 patients.
an = a + (n − 1)d
Write the formula for the nth term.
= 240 + (12 − 1)10
Substitute.
= 240 + 110 = 350
Simplify.
You will earn $350 if you visit 12 patients in 1 week.
Exercises
8. A boy starts a savings account for a mountain bike. He initially deposits $15. He decides
to increase each deposit by $8. How much is his 17th deposit?
10. Joe started a 30-min workout program this week. He wants to increase the workout
by 5 min every week. How long will his program be in the 16th week?
9-4
Summation notation shows the upper limit, lower limit, and explicit formula for the terms of a series.
To find the sum of an arithmetic series written in summation notation:

list the terms and add them, or use the formula sn 

n
a  an
2 1

15
  4n  1
n1
Use the formula sn 
n
 a1  an  .
2
a1 = 4(1) − 1 = 3
an = a15 = 4(15) − 1 = 59
sn 
15
2
(3  59)
First, find n, a1, and an. The upper limit is 15.
Evaluate the explicit formula at n = 1.
Evaluate the explicit formula at n = 15.
Substitute n = 15, a1 = 3, and an = 59.
= 465
Simplify.
The sum of the series is 465.
Exercises
Find the sum of each finite series.
3
4 1
1.   n  4 
2.  n
n1
n 1 3
9
5. 
n3
 4  2n 
5
6.  8n
n 1
8
3. 
n3
 3n  1
7
7.  4n
n2
8 2n
4. 
n 3 3
7
8.   3  2 n 
n 1
The debate club is offering a prize at the end of 10 weeks to a current member who brings
three new members for the first meeting, and then increases the number of new members
they bring each week by two thereafter. One member qualified for the prize with the
minimum number of new members. How many new members did the member bring at
Week 10? For all 10 weeks?
Step 1 Identify key information in the problem.
To win the prize, a member must bring three members to the first meeting, so a = 3. A
member must also bring two more new members to each meeting, so d = 2.
The contest extends for 10 weeks, so n = 10.
Step 2 Identify the information you are trying to find.
You want to find the 10th term, a10, and the sum of the first 10 terms, S10.
Step 3 Use the explicit formula to find a10.
an = a + (n − 1)d
Write the explicit formula.
a10 = 3 + (10 − 1)2
Substitute a = 3, d = 2, and n = 10.
a10 = 21
Simplify.
To win the prize, a member brought 21 new members to a meeting at Week 10.
Step 4 Use the value of a10 to find the total number of new members brought by the winner.
sn 
n
2
 a1  an 
Write the formula for the sum of an arithmetic series.
10
s10   3  21
2
Substitute a1 = 3, a10 = 21, and n = 10.
s10 = 120
Simplify.
The debate club had 120 new members brought in by the winner of the contest.
Exercises
9. The seating arrangement for a recital uses 20 seats in the first row and two additional
seats in each row thereafter. How many seats will be in the eighth row? In the ninth
row? How many seats total are there in the first nine rows?