Finding General Term of Squence

A sequence is a function whose domain is an ordered list of numbers. These numbers are positive integers starting with 1. Sometimes, people mistakenly use the terms series and sequence. A sequence is a set of positive integers while series is the sum of these positive integers. The denotation for the terms in a sequence is:

a1, a2, a3, a4, an, . . .

Finding the nth term of a sequence is easy given a general equation. But doing it the other way around is a struggle. Finding a general equation for a given sequence requires a lot of thinking and practice but, learning the specific rule guides you in discovering the general equation. In this article, you will learn how to induce the patterns of sequences and write the general term when given the first few terms. There is a step-by-step guide for you to follow and understand the process and provide you with clear and correct computations.

An arithmetic series is a series of ordered numbers with a constant difference. In an arithmetic sequence, you will observe that each pair of consecutive terms differs by the same amount. For example, here are the first five terms of the series.

3, 8, 13, 18, 23

Do you notice a special pattern? It is obvious that each number after the first is five more than the preceding term. Meaning, the common difference of the sequence is five. Usually, the formula for the nth term of an arithmetic sequence whose first term is a1 and whose common difference is d is displayed below.

an = a1 + (n - 1) d

Source: teacherspayteachers.com

Steps in Finding the General Formula of Arithmetic and Geometric Sequences
1. Create a table with headings n and an where n denotes the set of consecutive positive integers, and an represents the term corresponding to the positive integers. You may pick only the first five terms of the sequence. For example, tabulate the series 5, 10, 15, 20, 25, . . .

2. Solve the first common difference of a. Consider the solution as a tree diagram. There are two conditions for this step. This process applies only to sequences whose nature are either linear or quadratic.

Condition 1: If the first common difference is a constant, use the linear equation ax + b = 0 in finding the general term of the sequence.

a. Pick two pairs of numbers from the table and form two equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.

b. After forming the two equations, calculate a and b using the subtraction method.

c. Substitute a and b to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.

Condition 2: If the first difference is not constant and the second difference is constant, use the quadratic equation

a. Pick three pairs of numbers from the table and form three equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.


b. After forming the three equations, calculate a, b, and c using the subtraction method.

c. Substitute a, b, and c to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.




This article was originally published on The owlcation. Read the original article.


Source:
https://owlcation.com/stem/How-to-Find-the-General-Term-of-Arithmetic-and-Geometric-Sequences

Definition of Set

A set is a collection of objects, things or symbols which are clearly defined.
The individual objects in a set are called the members or elements of the set.

The following table shows some Set Theory Symbols.

A set must be properly defined so that we can find out whether an object is a member of the set.

1. Listing the elements (Roster Method)
The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

Example:
B = {2, 4, 6, 8, 10}
X = {a, b, c, d, e}

However, in some instances, it may not be possible to list all the elements of a set. In such cases, we could define the set by method 2.

2. Describing the elements
The set can be defined, where possible, by describing the elements. This is called the set-builder notation.

Example:
C = {x : x is an integer, x > – 3 }
This is read as: “C is the set of elements x such that x is an integer greater than –3.”

D= {x: x is a river in a state}

We should describe a certain property which all the elements x, in a set, have in common so that we can know whether a particular thing belongs to the set.

We relate a member and a set using the symbol ∈. If an object x is an element of set A, we write x ∈ A. If an object z is not an element of set A, we write z ∉ A.

∈ denotes “is an element of’ or “is a member of” or “belongs to”

∉ denotes “is not an element of” or “is not a member of” or “does not belong to”

Example:
If A = {1, 3, 5} then 1 ∈ A and 2 ∉ A

Basic vocabulary used in set theory

A set is a collection of distinct objects. The objects can be called elements or members of the set.
A set does not list an element more than once since an element is either a member of the set or it is not.

There are three main ways to identify a set:
1. A written description
2. List or Roster method
3. Set builder Notation

The empty set or null set is the set that has no elements.
The cardinality or cardinal number of a set is the number of elements in a set.
Two sets are equivalent if they contain the same number of elements.
Two sets are equal if they contain the exact same elements although their order can be different.

Definition and notation used for subsets and proper subsets

If every member of set A is also a member of set B, then A is a subset of B, we write A ⊆ B. We can also say A is contained in B.

If A is a subset of B, but A is not equal B then A is a proper subset of B, we write A ⊂ B.

The empty set is a subset of any set.
If a set A has n elements that it has 2n subsets.

How to use Venn diagrams to show relationship between sets and set operations?

A Venn diagram is a visual diagram that shows the relationship of sets with one another.
The set of all elements being considered is called the universal set (U) and is represented by a rectangle.

The complement of A, A', is the set of elements in U that is not in A.
Sets are disjoint if they do not share any elements.
The intersection of A and B is the set of elements in both set A and set B.
The union of A and B is the set of elements in either set A or set B or both.



This article was originally published on The online math learning. Read the original article.

Source:
https://www.onlinemathlearning.com/describing-sets.html