Arithmetic Sequences and Sums
Sequence
A Sequence is a set of things (usually numbers) that are in order.
Arithmetic Sequence
In an Arithmetic Sequence the difference between one term and the next is a constant.
In other words, you just add some value each time ... on to infinity.
Example:
| 1, 4, 7, 10, 13, 16, 19, 22, 25, ... |
This sequence has a difference of 3 between each number.
In General you could write an arithmetic sequence like this:
{a, a+b, a+2b, a+3b, ... }
where:
- a is the first term, and
- b is the difference between the terms (called the "common difference")
And you can make the rule by:
xn = a + b(n-1)
(We use "n-1" because b is not used in the 1st term).
Example: Write the Rule, and calculate the 4th term for
| 3, 8, 13, 18, 23, 28, 33, 38, ... |
This sequence has a difference of 5 between each number.
The values of a and b are:
- a = 3 (the first term)
- b = 5
(the "common difference")
The Rule can be calculated:
xn = a + b(n-1)
= 3 + 5(n-1)
= 3 + 5n - 5
= 5n - 2
So, the 4th term is:
x4 = 5·4 - 2 = 18
Is that right?
Summing an Arithmetic Series
To sum up the terms of an arithmetic sequence like this:
a + (a+b) + (a+2b) + (a+3b) + ...
you can use this formula:
What is that funny symbol? It is called Sigma Notation
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This symbol (called Sigma) means "sum up" |
And below and above it are shown the starting and ending values:
It says "Sum up n where n goes from 1 to 4. Answer=10
Here is how to use it:
Example: Add up the first 10 terms of the arithmetic sequence:
{ 1, 4, 7, 10, 13, ... }
The values of a, b and n are:
- a = 1 (the first term)
- b = 3 (the "common difference" between terms)
- n = 10 (how many terms to add up)
So:
Becomes:
 :
= 5(2+9·3) = 5(29) = 145
Check: why don't you add up the terms yourself, and see if it comes to 145
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