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 an activity.

The arithmetic series formula will make sense if you understand this activity. Focus then a lot on this activity!

Using the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Add the first and last terms of the sequence and write down the answer.

Then, add the second and next-to-last 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
Sum =

10
2

× (1 + 10)
We can make a

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.

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} =
_{1} + a_{n} )

n
2

× (an is the number of term, a

You will have no problem now to find the sum of 1 + 2 + 3 + 4 + ... + 100.

n = 100, a

S_{n} =

100
2

× (1 + 100 )
S

Find the sum of the arithmetic series 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50

n = 10, a

S_{n} =

10
2

× ( 5 + 50 )
S

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

When looking for the sum of arithmetic series, it is not always easy to know the number of terms or n.

Just use the formula below to find n.

last term - first term
common difference

+ 1
The common difference is the

How many term here? 2 + 6 + 10 + 14 + ... + 78

Common difference is 4

Number of terms =

78 - 2
4

+ 1 = 19 + 1 = 20 terms
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

$$ S_n = \sum_{i=1}^{10} 5n $$

Find the sum of arithmetic series with the quiz below: |