Sum of arithmetic series
Before I show you how to find the sum of arithmetic series, you need to know what an arithmetic series is or how to recognize it.
A series is an expression for the sum of the terms of a sequence.
For example, 6 + 9 + 12 + 15 + 18 is a series for it is the expression for the sum of the terms of the sequence 6, 9, 12, 15, 18.
By the same token, 1 + 2 + 3 + .....100 is a series for it is an expression for the sum of the terms of the sequence 1, 2, 3, ......100
To find the sum of arithmetic series, we can start with a an activity. The arithmetic series formula will make sense if you understand this activity. Focus then a lot on this activity!
Activity
Using the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Add the first and last terms of the sequence amd write down the answer
Then, add the second and nexttolast terms
Continue with the pattern until there is nothing to add
We get:
1 + 10 = 11
2 + 9 = 11
3 + 8 = 11
4 + 7 = 11
5 + 6 = 11
What patterns do see?
The sum is always 11
11 + 11 + 11 + 11 + 11 = 5 × 11 = 55
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
As you can see instead of adding all the terms in the sequence, you can just do 5 × 11 since you will get the same answer
Notice also that 5 × 11 =
10
/
2
× 11
We can make a
generalization that will help us find the sum of arithmetic series
Notice that 1 is the first term of the sequence. Notice also that 10 is the last term of the sequence
Sum =
10
/
2
× ( first term + last term)
For
10
/
2
,10 is the number of terms in the sequence since the sequence has 10 terms
Arithmetic series formula:
Sum =
number of terms
/
2
× (first term + last term)
The following notation is more commonly used to find the sum of arithmetic series
The sum S
_{n} of a
_{1} + a
_{2} + a
_{3} + a
_{4} + ... + a
_{n} is S
_{n} =
n
/
2
× (a
_{1} + a
_{n} )
n is the number of term, a
_{1} is the first term, and a
_{n} is the nth or last term
You will have no problem now to find the sum of 1 + 2 + 3 + 4 + ... + 100
n = 100, a
_{1} = 1, a
_{n} = 100
S
_{n} =
100
/
2
× (1 + 100 )
S
_{n} = 50 × 101 = 5050
Find the sum of the arithmetic series 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50
n = 10, a
_{1} = 5, a
_{n} = 50
S
_{n} =
10
/
2
× ( 5 + 50 )
S
_{n} = 5 × 55 = 275
Observation:
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = 5 × (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
We already found the sum of 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 above. It is 55.
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = 5 × 55 = 275
Number of terms:
When looking for the sum of arithmetic series, it is not always easy to know the number of terms
Just use this formula:
last term  first term
/
common difference
+ 1
The common difference is the
same number that is added to each term
How many term here? 2 + 6 + 10 + 14 + ... + 78
Common difference is 4
Number of terms =
78  2
/
4
+ 1 = 19 + 1 = 20 terms
Summation Notation:
See the summation notation for the series 8 + 14 + 20 + 26 + 32 + 38
If you are having hard time to derive the explicit formula, review
arithmetic sequence.
The technique is explained in arithmetic sequence
As you can see when
n = 1, 6 ×1 + 2 = 6 + 2 = 8
n = 2, 6 ×2 + 2 = 12 + 2 = 14
n = 3, 6 ×3 + 2 = 18 + 2 = 20
n = 4, 6 ×4 + 2 = 24 + 2 = 26
n = 5, 6 ×5 + 2 = 30 + 2 = 32
n = 6, 6 ×6 + 2 = 36 + 2 = 38
The big greek letter that looks like an E is the greek capital letter sigma. It is the equivalent of the English letter S for summation
Find the summation notation for 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50
A good observation may help you see that 5n is the explicit formula for 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Why? when n = 1, 5 × 1 = 5, when n = 2, 5 × 2 = 10, and so forth...
The upper limit is 10 since we have 10 terms
The lower limit is 1
S_{n} =
10
∑
n=1
5n
Find the sum of arithmetic series with the quiz below:


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