Arithmetic is one of the oldest elementary branches of the mathematics that is originated from the Greek word. This branch of mathematics involves the study of numbers, and the properties of the study of traditional operations like sum, difference, division, or multiplication etc. There are four basic operations applied to the arithmetic numbers and the advanced operations may include – logarithm values, exponential values, square roots etc. The concept of Arithmetic was started in the year 1801 and still valid in the practical world.

The ordered list of number is named as sequence and when they are added together, it is named as the series. The Arithmetic sequence is the sequence where each term is obtained or created by adding, subtracting the common numbers with its preceding value or terms. However, the differences among the adjacent terms would be the same in case of the arithmetic sequence. The Arithmetic sequence formula in mathematics is given as below –

\[\ a_{n}=a_{1}+(n-1)d\]

Where,

a_{n} – n^{th} term that has to be found

a_{1} – 1^{st} term in the seriesn

n- number of terms

d – common difference

The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include – (a, a + d, a + 2d, …. and so on) where a is the first term, d is the common difference between terms. There are two popular techniques to calculate the sum of an Arithmetic sequence. The formulas for both of the techniques are given below.

\[\ S=\frac{n}{2}(a+L)\]

\[\ S=\frac{n}{2}\left\{2a+(n-1)d\right\}\]

Where S is the sum of an Arithmetic sequence, a is the first term, d is the difference between two terms, L is the first term, and n is the total number of terms in the series.

Recursion is the process of choosing a beginning term and repeatedly apply the process to each term to reach the following term. With the help of Recursion, the value can be calculated quickly before the term you try to find out.

When talking about the Arithmetic Sequence Recursive Formula, it is divided into 2 parts – the first part defines the value beginning with the sequence and a recursion equation in the second part that shows how terms are related to the each other in respect to the preceding terms. The Arithmetic Sequence Recursive Formula in mathematics is given as below –

\[\ Arithmetic\;Sequence\;Recursive = t_{n} = t_{n-1}\]

With this formula, two or more values can be listed based on the nature of the sequence. However, the portion of the series is always dependent on the previous two terms are given in the sequence.

For direct computation of a term, you can use Arithmetic Sequence Explicit formula in mathematics. This formula is generally dedicated to the nth term if the sequence. All sequences are different when few are defined value, others have random numbers only. The Arithmetic Sequence Explicit Formula in mathematics can be given as –

\[\ Arithmetic\;Sequence\;Explicit = a_{n} = a_{1}+(n-1)d\]

Where, a_{n} is the nth term in the sequence, a_{1} is the first term in the series, n is the total number of terms, d is a common difference.