SequenceA Sequence is a set of things (usually numbers) that are in order. Show
Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for more details. Arithmetic SequenceIn 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:
Example: (continued)1, 4, 7, 10, 13, 16, 19, 22, 25, ... Has:
And we get: {a, a+d, a+2d, a+3d, ... } {1, 1+3, 1+2×3, 1+3×3, ... } {1, 4, 7, 10, ... } RuleWe 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 a rule, and calculate the 9th term, for this Arithmetic Sequence:
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:
Using the Arithmetic Sequence rule: xn = a + d(n−1) = 3 + 5(n−1) = 3 + 5n − 5 = 5n − 2 So the 9th term is: x9 = 5×9 − 2 Is that right? Check for yourself! Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s) Advanced Topic: Summing an Arithmetic SeriesTo 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
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:
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 Footnote: 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:
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: What are the 5 types of sequence?Sequences. 1Arithmetic sequences. ... . 2Geometric sequences. ... . 3Quadratic sequences. ... . 4Special sequences. ... . 1nth term of a linear sequence. ... . 2nth term of a quadratic sequence. ... . 3Use the nth term to calculate any term in a sequence. ... . 4Use the nth term to work out whether a number is in a sequence.. What is arithmetic and example?The definition of arithmetic refers to working with numbers by doing addition, subtraction, multiplication, and division. An example of arithmetic is adding two and two together to make four.
What are the 4 types of sequence?The types of sequence are:. Arithmetic sequence.. Geometric sequence.. Harmonic sequence.. Fibonacci sequence.. What are examples of arithmetic numbers?For example, the sequence 1, 6, 11, 16, … is an arithmetic sequence because there is a pattern where each number is obtained by adding 5 to its previous term.. 5, 8, 11, 14, .... 80, 75, 70, 65, 60, .... π/2, π, 3π/2, 2π, ..... -√2, -2√2, -3√2, -4√2, .... |