# Sigma Notation

## Sigma Notation - How it Works - Video

### Theorem 1 Sigma Notation:

m

Σ ak = the first term + the second term + the third term + ... + until the m term

k=1

Theorem on Sum of a Constant:

1 - The first statement means that the sum of the sequence is m times the constant if k is 1.

2 - The second statement means that the sum of the sequence is the quantity of m minus p plus times the constant if k is not 1.

### Theorem 2 Theorem 2:

1 - Statement 1 means that we can find the sum of the first sequence and then the second sequence and then combine them to find the sum of both.

2 - Statement 2 means that we can find the sum of the first sequence and then the second sequence and then subtract them to find the difference.

3 - Statement 3 means that we can find the sum of the sequence by factoring out the constant and then find the sum and then multiply by the constant to find the sum.

### Example 1 Example 1:

4

Σ 5

k=1

We can use either use the first Theorem on Sum of a Constant or the second to find the sum.

Using the first one, our constant is 5 and the last term we want is 4. Since k = 1, we can just multiply by 4 to get 20 to find the sum. If we write down a1 = 5, a2 = 5, a3 = 5, and a4 = 5, and add them, our result is 20.

Using the second one, we have the last term we want 4 and subtract out starting term in this case 1 and add 1 back and the result is 4. Since subtraction measures the distance between two numbers, we have add back 1 to find the total amount of numbers between any two numbers. For instance if we have the set of {2 - 6}, we don't have 6 - 2 = 4 numbers, we have 5 numbers, 2, 3, 4, 5, and 6.

Then we multiply by the constant, 5, and our result is 20.

### Example 2 Example 2:

6

Σ 5

k=4

Here, we can't use the first theorem since k does not equal 1, so we have to use the second one theorem.

So using the second one the, we have the last term we want 6 and subtract out starting term in this case 4 and add 1 back and the result is 3. Since subtraction measures the distance between two numbers, we have add back 1 to find the total amount of numbers between any two numbers. For instance if we have the set of {2 - 6}, we don't have 6 - 2 = 4 numbers, we have 5 numbers, 2, 3, 4, 5, and 6.

Then we multiply by the constant, 5, and our result is 15.

Or if we write down a4 = 5, a5 = 5, and a6 = 5, and add them, our result is 15.

### Example 3 Example 3:

3

Σ 2k + 4

k=1

Since we have small numbers, we can just plug in 1, 2, and 3 to find a1, a2, and a3. After find that a1 = 6, a5 = 8, and a6 = 10, we can add them together and our result is 24.

Now, if the numbers get bigger and bigger or we have more terms, adding each term is going to take forever. Let's take a look at what happens when we separate each sequence.

### Example 4 Example 4:

Our original is

3

Σ 2k + 4

k=1

So we can separate using Theorem on Sums - 1

3

Σ 2k

k=1

+

3

Σ 4

k=1

So we can separate the first part using Theorem on Sums - 3 by moving the constant outside the sigma notation.

2

*

3

Σ k

k=1

+

3

Σ 4

k=1

Now, we can use the different theorems to find the sum of the sequence.

2 * [ 1 + 2 +3 ] + 3 * 4

2 * 6 + 3 * 4

12 + 12

24

Write down each term