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Answer Key
Name: ____________________________________
Date: __________________
ARITHMETIC SERIES
COMMON CORE ALGEBRA II
A series is simply the sum of the terms of a sequence. The fundamental definition/notion of a series is below.
THE DEFINITION OF A SERIES
If the set a1 , a2 , a3 ,...  represent the elements of a sequence then the series, Sn , is defined by:
n
Sn   ai
i 1
In truth, you have already worked extensively with series in previous lessons almost anytime you evaluated a
summation problem.
Exercise #1: Given the arithmetic sequence defined by a1  2 and an  an 1  5 , then which of the following is
5
the value of S5   ai ?
a2  a1  5  2  5  3
a3  a2  5  3  5  8
i 1
(1) 32
(3) 25
(2) 40
(4) 27
a4  a3  5  8  5  13
(2)
a5  a4  5  13  5  18
S5  a1  a2  a3  a4  a5  40
The sums associated with arithmetic sequences, known as arithmetic series, have interesting properties, many
applications and values that can be predicted with what is commonly known as rainbow addition.
Exercise #2: Consider the arithmetic sequence defined by a1  3 and an  an 1  2 . The series, based on the
first eight terms of this sequence, is shown below. Terms have been paired off as shown.
(a) What does each of the paired off sums equal?
Each pair has a sum of 20.
(b) Why does it make sense that this sum is constant?
It makes sense that this sum is constant because as
one term increases by 2, the other decreases by 2.
(c) How many of these pairs are there?
3  5  7  9  11  13  15  17
(d) Using your answers to (a) and (c) find the value
of the sum using a multiplicative process.
The number of pairs will always be half the total
number of elements, in this case 4 pairs.
4  20   80
(e) Generalize this now and create a formula for an arithmetic series sum based only on its first term, a1 , its last
term, an , and the number of terms, n.
In general, each pair will have the sum of the first and last terms, a1  an . The number of pairs will always be half the
number of total elements, or n . So, our sum will be given by S n  n  a1  an  .
2
2
COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
SUM OF AN ARITHMETIC SERIES
Given an arithmetic series with n terms, a1 , a2 , ..., an  , then its sum is given by:
Sn 
n
 a1  an 
2
Exercise #3: Which of the following is the sum of the first 100 natural numbers? Show the process that leads to
your choice.
(1) 5,000
S100  1  2  3      100
(3) 10,000
(2) 5,100
100

 1  100    50  101  5050
2
(4) 5,050
(4)
Exercise #4: Find the sum of each arithmetic series described or shown below.
(b) The first term is 8 , the common difference, d,
is 6 and there are 20 terms
(a) The sum of the sixteen terms given by:
10  6  2    46  50 .
S16 
a20  8  19  6   106
16
  10  50 
2
S 20 
 8  40  320
20
  8  106 
2
 10  98   980
(c) The last term is a12  29 and the common
difference, d, is 3 .
(d) The sum 5  8  11    77 .
77  5  3  n  1   77  5  3n  3
29  a1  11  3   29  a1  33  a1  4
77  3n  2  3n  75  n  25
12
S12 
 4  29 
2
S 25 
S12  6  25   150
25
 5  77   1025
2
Exercise #5: Kirk has set up a college savings account for his son, Maxwell. If Kirk deposits $100 per month
in an account, increasing the amount he deposits by $10 per month each month, then how much will be in the
account after 10 years?
a120  100  119  10   1290
S120 
120
 100  1290   60  1390 
2
 $83, 400
COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
Answer Key
Name: ____________________________________
Date: __________________
ARITHMETIC SERIES
COMMON CORE ALGEBRA II HOMEWORK
FLUENCY
1. Which of the following represents the sum of 3  10    87  94 if the arithmetic series has 14 terms?
(1) 1,358
(3) 679
(2) 658
(4) 1,276
S14 
14
 3  94   7  97   679
2
(3)
2. The sum of the first 50 natural numbers is
(1) 1,275
(3) 1,250
(2) 1,875
(4) 950
S 50  1  2  3      49  50

50
 1  50   25  51  1275
2
(1)
3. If the first and last terms of an arithmetic series are 5 and 27, respectively, and the series has a sum 192,
then the number of terms in the series is
(1) 18
(3) 14
(2) 11
(4) 12
192 
n
n
 5  27   192   32 
2
2
(4)
16n  192  n  12
4. Find the sum of each arithmetic series described or shown below.
(a) The sum of the first 100 even, natural
numbers.
S100  2  4  6      200

100
 2  200   50  202 
2
 10,100
(c) A series whose first two terms are
12 and  8 , respectively, and whose last
term is 124.
(b) The sum of multiples of five from 10 to 75,
inclusive.
75  10  5  n  1   75  10  5n  5
75  5n  5  5n  70  n  14
S14 
14
 10  75   7  85   595
2
(d) A series of 20 terms whose last term is equal
to 97 and whose common difference is five.
From the first two terms we can see that
the common difference is d  4 
124  12  4  n  1  124  12  4n  4
124  4n  16  4n  140  n  35 terms
S 35 
35
 12  124   1, 960
2
a20  a1  19d  97  a1  19  5 
97  a1  95  a1  2
S 20 
20
 2  97   990
2
COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
5. For an arithmetic series that sums to 1,485, it is known that the first term equals 6 and the last term equals
93. Algebraically determine the number of terms summed in this series.
1485 
n
n
 6  93   1485   99   99n  2970  n  30
2
2
APPLICATIONS
6. Arlington High School recently installed a new black-box theatre for local productions. They only had
room for 14 rows of seats, where the number of seats in each row constitutes an arithmetic sequence starting
with eight seats and increasing by two seats per row thereafter. How many seats are in the new black-box
theatre? Show the calculations that lead to your answer.
a14  a1  13d  8  13  2   34 seats in last row
S14 
14
 8  34   7  42   294 total seats
2
7. Simeon starts a retirement account where he will place $50 into the account on the first month and
increasing his deposit by $5 per month each month after. If he saves this way for the next 20 years, how
much will the account contain in principal?
a240  a1  239d  50  239  5   $1245 in the last month
S 240 
240
 50  1245   120  1295   $155, 400
2
8. The distance an object falls per second while only under the influence of gravity forms an arithmetic
sequence with it falling 16 feet in the first second, 48 feet in the second, 80 feet in the third, etcetera. What
is the total distance an object will fall in 10 seconds? Show the work that leads to your answer.
This is an arithmetic sequence whose terms are
separated by a common difference of 32. So, we
can find the last or 10th term:
10
d
i 1
i

10
 16  304   5  320   1, 600 ft
2
d10  d1  9  32   16  9  32   304
9. A large grandfather clock strikes its bell once at 1:00, twice at 2:00, three times at 3:00, etcetera. What is the
total number of times the bell will be struck in a day? Use an arithmetic series to help solve the problem and
show how you arrived at your answer.
We can start adding up the number of
times the clock strikes:
1  2  3      12  1  2  3    12
1  2  3      12 
12
 1  12   6 13   78
2
78  78  156 bell strikes per day
COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
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