11-4 Arithmetic Series
Drill
Find each sum.
1) 2 + 3.5 + 5 + 6.5 + 8
2) -17 + (-13) + (-9) + (-5) + (-1) + 3
Write an explicit formula for each sequence.
3) 4, 6, 8, 10, 12, …
4) 1, 4, 7, 10, 13, 16, …
5) -17, -23, -29, -35, …
Homework
None
6) 10, 1, -8, -17, …
11-4 Arithmetic Series
Algebra 2, Belllman, Bragg, Charles, Handlin, Kennedy, Prentice Hall
1. WRITING AND EVALUATING ARITHMETIC SERIES
VOCABULARY
Series: The expression for the sum of terms of a sequence.
Finite Sequences and Series: have terms that you can count from 1 to a final
whole number n.
Finite sequence: 1, 2, 4, 8
Finite series: 1 + 2 + 4 + 8
Infinite Sequences and Series: continue without end
Infinite sequence: 1, 2, 4, 8, …
Infinite series: 1 + 2 + 4 + 8 + …
Arithmetic Series: A series whose terms form an arithmetic sequence.
6, 12, 18
(arithmetic sequence – Add six)
6 + 12 + 18 = 36
Examples:
Write the related series for each finite sequence. Then evaluate the
series.
1.
2, 11, 20, 29, 38, 47.
Answer: 2 + 11 + 20 + 29 + 38 + 47 = 147
2. 100, 125, 150, 175, 200, 225
Answer: 100 + 125 + 150 + 175 + 200 + 225 = 975
Tell whether each list is a sequence or series. Then tell whether it is
finite or infinite.
3.
1, 0.5, 0.25, 0.125, 0.0625
sequence, finite
4.
-0.5 - 0.25 – 0.125- …
series, infinite
5.
2.3 + 4.6 + 9.2 + 18.4
series, finite
When you know only the first and last terms of an arithmetic sequence,
or if the sequence has many terms, you can use a formula to evaluate the
related series quickly.
Sum of a Finite Arithmetic Series:
The sum S n of a finite arithmetic series a1  a2  a3  ...  an is:
Sn 
n
(a1  a n )
2
Where n is the number of terms, a1 is the first term, and a n is the last
term.
Examples:
6.
Evaluate the sum for the related series of 8 terms:
1, -1, -3, …, -13
S8 
8
(1  (13))
2
= -48
7. Evaluate the sum: 10 + 7 + 4 + …; n = 5
10 + 7 + 4 + 1 + (-2) = 20
a n  a1  (n  1)d
a5  10  (5  1)( 3)
a5  2
8. 6 + 7.4 + 8.8 + …; n = 11
n
(a1  a n )
2
11
S11  (6  20)
2
S11  143
Sn 
a11  6  (11  1)(1.4)
a11  20
2. USING SUMMATION NOTATION
VOCABULARY
Sigma – Greek capital letter sigma

means summation
Limits – least and greatest integral values of n
Summation Notation
3
 (5n  1)
where 3 is the upper limit, 1 is the lower limit, and 5n +1 is
n 1
the explicit formula for the sequence
Evaluate as:
[5(1) + 1] + [5(2) + 1] + [5(3) + 1] = 6 + 11 + 16 = 33
You can use the summation symbol  to write a series.
Examples:
Use summation notation to write the following series.
9.
3 + 6 + 9 + … for 33 terms.
33
 3n
n 1
10. 1 + 2 + 3 + … for 6 terms
6
n
n 1
11. 3 + 8 + 13 + 18 + … for 9 terms
9
 (5n  2)
n 1
To expand a series from summation notation, you can substitute each
value of n into the explicit formula and add the results.
Examples:
For each sum, find the number of terms, the first term and the last
term. Then evaluate the series.
12.
10
 (n  3)
n 1
First term: 1 – 3 = -2
n
(a1  a n )
2
10
S10  (2  7)  25
2
Sn 
Last term: 10 – 3 = 7
13.
4 1
 ( n  1)
n 1 2
First term: 1.5
Last term: 3
S4  9
5
14.
n
2
n2
First term: 4
Last term: 25
S 4  58
Homework
p. 622 #1 – 29 (eoo)