What are the differences between arithmetic sequence, arithmetic mean and arithmetic series?

Basically arithmetic is a well-known branch of mathematics that deals with numbers, their properties and traditional operations on them. It is also referred to the arithmetic calculations i.e; the operations relating to them for real numbers involve division, multiplication, addition, subtraction and other than that the extraction of roots and exponentiation. Mathematics includes many concepts and there are many approaches to understanding these concepts and learning completely.

Some of the terms and subcategories that made the understanding and calculations easier are arithmetic sequence, arithmetic mean and arithmetic series. In this article, students will be able to learn the differences among the arithmetic sequence, arithmetic means and arithmetic series. 

What is Arithmetic Sequence?

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is always constant. The sequence that starts from any number is the result of the difference between consecutive terms which always are the same. This pattern involves the addition or subtraction of a number to each term in reference to getting the next term. 

For example, if you start a series by a number 2 then the arithmetic sequence will be obtained with the same difference of 2 i.e; 2, 4, 6, 8, 10, ….

Rule of Arithmetic sequence can be written as follows:

 x_n = a + d ( n−1 )

In this formula

1. Xn is the nth term of the sequence
2. a is the first term of the sequence.
3. d is the common difference.
4. n is the number of term 

To calculate you can also use an easy to use arithmetic sequence calculator for free to calculate quickly.

Arithmetic Mean

Arithmetic mean is referred to as the arithmetic average. It is calculated by the addition of all numbers in a given data and dividing them by a total number of items in a given data set. It is assumed that the arithmetic average for evenly distributed numbers is quite equal to the middle number of the given data set. The basic formula for the arithmetic means for the given data can be written as follows in simple words:

Sum of all values

Mean = ______________

Number of values

This formula for the arithmetic means elaborates that the sum of all the given values when divided by the total number of values then the arithmetic mean can be found. For example, if you have a number of values 1 +2+3+4+5 then what would the arithmetic mean?

Mean = 1+ 2+ 3+ 4+ 5 = 15 = 3

5     5

It means after the calculation through a simple mean formula you have the average or mean is 3.

Yeah the values can be more than that. And if you are facing too many values at the same time so to save time use a mean calculator online for free to calculate them quickly.

Arithmetic series

Arithmetic series refers to the sum of the terms in an arithmetic sequence with a definite number of terms in a given set of data. The arithmetic sequence often results from the addition of terms of an arithmetic sequence. 

The standard formula for the arithmetic series can be written as follows:

an = a1 + ( n-1 ) d

Whereas the description of this formula is:

1. n is the nᵗʰ term
2. a1 is the first term
3. d is the common difference between the terms of the sequence

Difference between arithmetic sequence, arithmetic mean and arithmetic series

The difference between the arithmetic sequence, arithmetic mean and arithmetic series can be explained as follows:

1. An arithmetic sequence is referred to the sequence where difference between the successive terms is constant while the arithmetic mean is known as the numbers between arithmetic extremes and the arithmetic series referred to as the sequence of the nth partial sums of the original arithmetic sequences. 
2. An arithmetic sequence written in the terms of the first term is a1, common difference  d and index n whereas the arithmetic series is simply the sum of all terms of the arithmetic sequence.
3. The arithmetic sequence has the importance of order of elements  while the arithmetic series does not.
4. Examples for arithmetics sequences fall in infinite sequence i.e; 1, 2, 3, and so on while the arithmetic series has the infinite series i.e; 1 +2 +3 +4 + and so on.
5. Arithmetic mean is the measure of the central tendency while the arithmetic sequence is the elements that follow the pattern.
6. The arithmetic mean is known as the simplest measure of mean or average while the arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. 
7. The arithmetic sequence is simply the elements that follow the pattern while the arithmetic series refers to the sum of elements of the series.