Writing Rules for Arithmetic Sequences | Study.com
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, we just add the same value each time ... infinitely.
Example:
1, 4, 7, 10, 13, 16, 19, 22, 25, ... |
This sequence has a difference of 3 between each number.
In General we could write an arithmetic sequence like this:
{a, a+d, a+2d, a+3d, ... }
where:
- a is the first term, and
- d is the difference between the terms (called the "common difference")
Example: (continued)
1, 4, 7, 10, 13, 16, 19, 22, 25, ... |
Has:
- a = 1 (the first term)
- d = 3 (the "common difference" between terms)
And we get:
{a, a+d, a+2d, a+3d, ... }
{1, 1+3, 1+2×3, 1+3×3, ... }
{1, 4, 7, 10, ... }
Rule
We can write an Arithmetic Sequence as a rule:
xn = a + d(n-1)
(We use "n-1" because d 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 d are:
- a = 3 (the first term)
- d = 5 (the "common difference")
The Rule can be calculated:
xn = a + d(n-1)
= 3 + 5(n-1)
= 3 + 5n - 5
= 5n - 2
So, the 4th term is:
x4 = 5×4 - 2 = 18
Is that right? Check for yourself!
Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s)
Summing an Arithmetic Series
To sum up the terms of this arithmetic sequence:
a + (a+d) + (a+2d) + (a+3d) + ...
use this formula:
What is that funny symbol? It is called Sigma Notation
(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, d and n are:
- a = 1 (the first term)
- d = 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
Why Does the Formula Work?
Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.
First, we will call the whole sum "S":
S = a + (a + d) + ... + (a + (n-2)d) + (a + (n-1)d)
Next, rewrite S in reverse order:
S = (a + (n-1)d) + (a + (n-2)d) + ... + (a + d) + a
Now add those two, term by term:
S | = | a | + | (a+d) | + | ... | + | (a + (n-2)d) | + | (a + (n-1)d) |
S | = | (a + (n-1)d) | + | (a + (n-2)d) | + | ... | + | (a + d) | + | a |
2S | = | (2a + (n-1)d) | + | (2a + (n-1)d) | + | ... | + | (2a + (n-1)d) | + | (2a + (n-1)d) |
Each term is the same! And there are "n" of them so ...
2S = n × (2a + (n-1)d)
Now, just divide by 2 and we get:
S = (n/2) × (2a + (n-1)d)
Which is our formula:
Comments
Monisoyoqevemo
18-05-2017 11:06 Comments
AUR Package Update: carl ( An Open Source C++ Library for Computer Arithmetic and Logic )
Pomoyalexamo
19-05-2017 12:23 Comments
Definition of taking for people for granted is telling people the majority want a second EUref when only 22% do. Diane Abbott arithmetic!
Venebazenubeb
12-06-2017 01:55 Comments
Physics professors who crucify you for arithmetic errors are not the type of physics professors you need in your life.
Xobakefeducos
27-06-2017 17:03 Comments
Big arithmetic is coming for you
Gikuhopicuxemo
10-07-2017 20:43 Comments
The esoteric number philosophy of -study until one sees the nature of numbers with the mind for the sake of the
Diwipowuyohoc
09-08-2017 21:35 Comments
ARITHMETIC FOR THE ARITHMETIC GOD! PROGRESSIONS FOR THE PROGRESSION MISTRESS
Sulasisobejava
14-11-2017 15:01 Comments
Physics professors who crucify you for arithmetic errors are not the type of physics professors you need in your life
Suquragekif
25-11-2017 17:47 Comments
The Irish language has one set of numbers for arithmetic, one set for humans and one for counting non-human.
Yolosayo
07-12-2017 10:54 Comments
Sonko does not rely on the tribal arithmetic leads a movement for the"POOR TRIBE" from Nairobi slums who are the majority.
Tizahoc
18-12-2017 05:37 Comments
NOT democratically elected! Why is simple arithmetic so hard for you?
Robehumuripige
27-01-2018 09:59 Comments
Pass ordering is not a problem for Prepack for concrete values since it"s just executed. Abstract arithmetic like this should be doable too.
Hekokusinoko
07-02-2018 05:22 Comments
Unlike TPU 1.0, it doesn"t do quantized integer arithmetic. Jeff Dean says you can use the same FP representation for training and inference