LESSON 7-4 : Arithmetic Series
Terminology
 Series – the sum of the terms of a
.

Arithmetic Series - the sum of the terms of an

Sn - the partial sum of the first n terms of a sequence where,
Sn = t1 + t2 + t3 + … + tn-1 + tn
The arithmetic series is:

sequence.
Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + [a + d(n – 2)] + [a + d(n – 1)]
The General Formula for Sn
How did Karl Friedrich Gauss find the sum of the first 100 natural numbers?
Solution:
S100 =
1+
2
+ 3 + . . . + 99 + 100
write in reverse: S100 = 100 + 99 + 98 + . . . + 2
+ 1
... (1)
...(2)
Add (1) + (2) : 2 S100= 101 + 101 + 101 +. . . + 101 + 101
2 S100= 100(101)
2
2
How many 101s?
100 of them!
S100 = 50(101)
S100 = 5050
How can you find the sum of any number of terms in an arithmetic sequence?
Solution:
Sn =
write in reverse: Sn =
a
+
(a + d) + (a + 2d) + . . . + (tn - d)
tn +
tn
(tn – d) + (tn – 2d) + . . . + (a + d) + a
Add (1) + (2) : 2 Sn = (a + tn) + (a + tn)+ (a + tn)+ . . .
2 S n=
2
+
...(1)
...(2)
+ (a + tn) + (a + tn)
n(a + tn)
2
How many (a+tn)?
n of them!
Sn = n(a + tn)
2
or
Sn = n(t1 + tn)
2
Also, since tn = a + d(n-1),
Substitute for tn :
Arithmetic Series given last term, tn
Sn = n(a + tn)
2
Sn = n[a + a + d(n-1)]
2
Sn = n[2a + d(n-1)]
2
Arithmetic Series given
common difference,
1. For the series 2 + 11 + 20 + 29 + … , find S20.
2. The fifth term of an arithmetic series is 9, and the sum of the first 16 terms is 480. Find
the first three terms of the series.
3. Find the sum of the arithmetic series if the series is 3+7+11+…+479.
4. A stadium has 25 rows. The 1st row has 40 seats. Each successive row has 1 more
seat. How many seats are there in the stadium?
5. Sukaina deposited $128 in her account. Each week, she deposits $7 less than the
previous week until she makes her last deposit of $9. Find the total value of her
deposits.
Homefun: p. 452 #2, 3, 5, 6bdf, 10, 11, 13, 15
LESSON 6: Arithmetic Series
Chapter 6: Sequences and Series
Arithmetic Series SOLUTIONS
Arithmetic Series


series – is the sum of the terms of a sequence.
The sum of the first n terms of a sequences is Sn, where,
Sn = t1 + t2 + t3 + … + tn-1 + tn

The arithmetic series is:
Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + [a + (n – 2)d] + [a + (n – 1)d]
Find the sum of the series Sn, by adding the first term, t1, and the last term, tn,
together, multiplying by the number of terms, n, and then dividing the result by 2.
n(t1  t 2 )
Sn 
2
Remember that t1 = a and tn = a + (n – 1)d, so we substitute this into the above
equation we get:
n
Sn  2a  (n  1)d 
2


Examples
1.
For the series 2 + 11 + 20 + 29 + … , find
a) t20
b) S20
a) t20 = 2 + 19(9) = 173
20
2(2)  (20  1)9
S2 0 
b)
2
 1750
2.
The fifth term of an arithmetic series is 9, and the sum of the first 16
terms is 480. Find the first three terms of the series.
t5 = 9
S16 =480
(1) 9 = a +(5 – 1)d  9 = a + 4d
(2) 480 = 16/2(2a + (16 – 1)d)  480 = 2a + 15d
When solving this system we get:
d=6
a = -15
.: -15, -9, -3
3.
Find the sum of the arithmetic series.
3 + 7 + 11 + … + 479
a=3
d=4
479 = 3 + (n – 1)4
479 = 3 – 4 + 4n
479 = 4n – 1
n = 120
120
2(3)  (120  1)4
2
 28920
S120 
4.
Carol deposited $128 in her account. Each week, she deposits $7 less
than the previous week until she makes her last deposit of $9. Find the total value of
her deposits.
7-4: Arithmetic Series
Arithmetic Series


series – is the sum of the terms of a sequence.
The sum of the first n terms of a sequences is Sn, where,
Sn = t1 + t2 + t3 + … + tn-1 + tn

The arithmetic series is:
Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + [a + (n – 2)d] + [a + (n – 1)d]
Find the sum of the series Sn, by adding the first term, t1, and the last term, tn,
together, multiplying by the number of terms, n, and then dividing the result by 2.
n (t 1  t n )
Sn 
2
Remember that t1 = a and tn = a + (n – 1)d, so we substitute this into the above
equation we get:
n
Sn  2a  (n  1)d 
2


Examples
5. For the series 2 + 11 + 20 + 29 + … , find
c) t20
d) S20
6. The fifth term of an arithmetic series is 9, and the sum of the first 16 terms is 480. Find
the first three terms of the series.
7. Find the sum of the arithmetic series.
3 + 7 + 11 + … + 479
8.
Carol deposited $128 in her account. Each week, she
deposits $7 less than the previous week until she makes her last deposit of $9. Find the
total value of her deposits.
Homefun: p. 452 #2, 3, 5, 6bdf, 10, 11, 13, 15