Arithmetic Series

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Arithmetic Series
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Definition of an arithmetic series.
The sum of the terms in an arithmetic sequence.
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Arithmetic Sequence
Arithmetic Series
4 7 10 13
4, 7, 10, 13
-10, -4, 2
-10 -4 2
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Example 1. During Kwanzaa one candle is lit the
first night, two on the second night, and so
forth for seven nights.
How many candles are lit in all.
1 2 3 4 5 6 7 28
The symbol Sn is used to represent the sum of the
first n terms of a series. The above represents
S7.
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Suppose we write S7 in two different orders and
find the sum
S7 1 2 3 4 5 6 7
S7 7 6 5 4 3 2 1
2S7 8 8 8 8 8 8 8
7 sums of 8
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S7 1 2 3 4 5 6 7
S7 7 6 5 4 3 2 1
2S7 8 8 8 8 8 8 8
7 sums of 8
2S7 7(8)
Now analyze this expression in terms of Sn.
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Now analyze this expression in terms of Sn.
7 represents n and 8 represents the sum of the
first and last terms, a1an.
Thus we can replace the expression with
8
This formula can be used to find the sum of any
arithmetic series.
Sum of an arithmetic series.
The sum of the first n terms of an arithmetic
series is given by
where n is a positive integer.
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Example 1. Find the sum of the first 50 positive
even integers.
25(102)
2550
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We have discovered in a previous lesson that an
a1 (n-1)d
Using this formula and substitution gives us
another version of the formula for the sum of an
arithmetic sequence.
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replace an
This formula is useful when we do not know the
value of the last term.
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Example 2.
Find the sum of the first 40 terms of an
arithmetic series in which a1 70 and d -21
The series is 70 49 27 ...
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Example 2. Sum of first 40 terms a1 70 and d
-21
The series is 70 49 27 ...
Sn
-13580
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Example 3. Physics
When an object is dropped it falls 16 feet in the
first second, 48 feet in the second second, and
80 feet in the third second.
How many feet would if fall in the 20th second.
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Example 3. Object falling 16, 48, and 80 feet in
1, 2, and 3 seconds.
How many feet would if fall in the 20th second.
16 48 80
32
32
Common difference is 32
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Example 3. Object falling 16, 48, and 80 feet in
1, 2, and 3 seconds.
How many feet would if fall in the 20th second. d
32
an a1 (n-1)d
a20 16 (20-1)32
a20 624
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Example 4. Refer to example 3
How many feet would a free falling object fall in
20 seconds?
S20 6400
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Example 5.
Find the first three terms of an arithmetic
series in which a1 13, an 157, and Sn 1445
We are given a1, an, and Sn.
Therefore we use the formula
and solve for n.
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Example 5. Find the first 3 terms if a1 13, an
157, and Sn 1445
solve for n.
1445
1445 85n
1445
n 17
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Example 5. Find the first 3 terms if a1 13, an
157, and Sn 1445 n 17, now find d.
an a1 (n-1)d
157 13 (17-1)d
157 13 16d
144 16d
d 9
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Example 5. Find the first 3 terms if a1 13, an
157, and Sn 1445 n 17, d 9.
a2 a1 d
a3 a2 d
a3 22 9
a2 13 9
a3 31
a2 22
The first 3 terms are 13, 22, and 31.
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To simplify writing out series we use sigma or
summation notation.
2 4 6 8 ... 20 is written
This expression is read
the sum of 2n as n increases from one to ten.
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Last value of n
First value of n
Formula for the seq
The variable below the ? sigma is called the
index of summation. The upper limit is the upper
limit of the index.
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The variable below the ? sigma is called the
index of summation. The upper limit is the upper
limit of the index.
To generate the terms of the series,
successively replace the index of summation
with consecutive integers as values of n. In
this series n 1,2,3, and so on, through 10.
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Example 6.
Write the terms of
and find the sum.
(235) (245) (255) (265) (275)

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