 # Arithmetic Sequence Explicit Formula

The Arithmetic Sequence Explicit formula allows the direct computation of any term for an arithmetic sequence. In mathematical words, the explicit formula of an arithmetic sequence is designated to the nth term of the sequence.

A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is called a1, the common difference is d, and the number of terms is n. While some sequences are simply random values, while other sequences have a definite pattern that is used to arrive at the sequence’s terms.

At BYJU’S you will get to know the formula of Arithmetic Sequence Explicit and few solved examples that will help you to understand this mathematical formula.

Here is the formula of Arithmetic Sequence Explicit: an = a1 + (n – 1)d

Where,

$$\begin{array}{l}a_{n}\end{array}$$
is the
$$\begin{array}{l}n^{th}\end{array}$$
term in the sequence
$$\begin{array}{l}a_{1}\end{array}$$
is the first term in the sequence
n is the term number
d is the common difference

### Examples of Arithmetic Sequence Explicit formula

Example 1: Find the explicit formula of the sequence 3, 7, 11, 15, 19…

Solution:
The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields 4.
Therefore:

$$\begin{array}{l}a_{1}\end{array}$$
= 3
d = 4

And the explicit formula is

$$\begin{array}{l}a_{n}\end{array}$$
= 3 + (n – 1)4

= 3 + 4n – 4

= 4n – 1

Example 2: Find the explicit formula of the sequence 3, -2, -7. -12…

Solution:
The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields -5.
Therefore:

$$\begin{array}{l}a_{1}\end{array}$$
= 3
d = -5

And the explicit formula is

$$\begin{array}{l}a_{n}\end{array}$$
= 3 + (n – 1)(–5)

= 3 – 5n + 5

= 8 – 5n