# 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 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 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 Sn of a1 + a2 + a3 + a4 + ... + an   is  Sn =
n / 2
× (a1 + an )

n is the number of term, a1 is the first term, and an is the nth or last term

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

n = 100, a1 = 1, an = 100

Sn =
100 / 2
× (1 + 100 )

Sn = 50 × 101 = 5050

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

n = 10, a1 = 5, an = 50

Sn =
10 / 2
× ( 5 + 50 )

Sn = 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

Sn = 10 n=1 5n

 Find the sum of arithmetic series with the quiz below:

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