# Study of Arithmetical Sequence and Their Types

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An arithmetic sequence is always confused with an arithmetic series. There is a difference between arithmetic sequence and arithmetic series. An arithmetic sequence is an elementary topic and is widely used by students while calculating.

You can define a sequence as an orderly arrangement of objects. In Arithmetic, the term arithmetic sequence is called a sequential collection of elements repeated orderly.

The arithmetic sequence is also known as arithmetic progression. The repetitions can be e of any sort. The arithmetic series is the sum of all the elements.

You can also define an arithmetic sequence as a list of sequential items and objects where is arithmetic series is the total of objects sequentially.

The arithmetic sequence is the study of the fundamentals. You have to solve problems after learning the formulas. In the case of the arithmetic sequence, the individual terms are in repetition at various positions.

Based on the length of the sequence, the Arithmetic sequence is of two types

1. Finite arithmetic sequence

2. Infinite arithmetic sequence

The length of the sequence depends on the number of terms occurring in a definite formula.

In Arithmetic sequence, the arrangement of any item or set of numbers is in a repetitive and particular order that follows a rule.

If x1, X2, X3, and X4 represent a sequence then the numeric value 1, 2, 3, and 4 denotes the position of the terms.

In an Arithmetic sequence, the terms are arranged in a finite sequence or an infinite sequence.

But in the case of arithmetic series, if X1 and X2 are sequential numbers, then the corresponding series would be SN is equal to A1 plus A2.

## There are four types of sequences-

1. Arithmetic sequence
2. Geometric sequence
3. Harmonic sequence
4. Fibonacci numbers

1. Arithmetic sequence- In this type of arithmetic sequence the term is made by adding or subtracting a definite number from the preceding number.

2. Geometric sequence- Geometric sequence is obtained by multiplying or dividing a definite number with the preceding number

3. Harmonic sequence- In this type of sequence, the reciprocal of all the numbers of the sequence form an Arithmetic sequence.

4. Fibonacci numbers- This is an important sequence where each element is obtained by adding the preceding element with the successive element and the sequence begins with 0 and 1.

Example of Fibonacci numbers is – f0 is equal to zero and F1 where Fn is equal to Fn + 1 + Fn – 2

### Formulas for an arithmetic sequence are as follows

1. Sequence – It is equal to a, a plus d, a plus 2d…… a + [n- 1) d…….

2. Common difference or ratio – In this, you subtract the preceding term from the successive term, d= A2- A1

3. General term with an nth term- an a+ (n-1)

### Formulas for a geometric sequence are as follows-

1. a, at, ar square equal to ar (n- 1)

2. Common difference or ratio- in this the preceding term is divided by a successive term.

3. General Term with nth term- an = 1\r (n-1)

In these formulas, a stands for the first term d stands for common difference n stands for the position of the term and r stands for common ratio.

Cuemath is an online training program where you develop your arithmetic skills with constant practice and the guidance of the teacher.

Cuemath classes are beneficial for the student who wants to master this subject for further exams.

#### There are four differences between arithmetic sequence and arithmetic series-

1. In an Arithmetic sequence the set of elements follows a definite pattern whereas in an Arithmetic series, the elements are added sequentially.

2. In the Arithmetic sequence the order and the sequence of elements are very important but in the Arithmetic series, the order of elements is not important.

3. Infinite arithmetic sequence elements are denoted by 1, 2, 3 whereas in arithmetic series elements are denoted by 1+ 2+ 3.

4. In infinite arithmetic sequence the elements are denoted by 1, 2, 3……….. Whereas in infinite arithmetic series, the elements are denoted by 1 + 2+ 3 ……..

An arithmetic sequence is not a difficult topic. It could be well understood if you learn the formulas and use them in your calculations.