# Arithmetic Sequence Explicit Formula

Before going to learn the arithmetic sequence explicit formula, let us recall what is an arithmetic sequence. It is a sequence of numbers in which the differences between every two consecutive numbers are all same. For example, -3, -6, -9, -12, ... is an arithmetic sequence as the difference between every term and its previous term is -3. This difference is usually known as the common difference and is denoted by d. The first term of an arithmetic sequence is represented by \(a_1\) or a. Let us learn the arithmetic sequence explicit formula along with a few solved examples.

## What Is Arithmetic Sequence Explicit Formula?

The arithmetic sequence explicit formula is used to find any term (n^{th} term) of the arithmetic sequence, \(a_1, a_2, a_3, ..., a_n,....\) using its first term (a) and the common difference (d). This formula gives the n^{th} term formula of an arithmetic sequence. The arithmetic sequence explicit formula is:

### Arithmetic Sequence Explicit Formula

The arithmetic sequence explicit formula is:

\(a_n\) = a + (n - 1) d

where,

- \(a_n\) = n
^{th}term of the arithmetic sequence - a = the first term of the arithmetic sequence
- d = the common difference (the difference between every term and its previous term. i.e., d \(=a_n-a_{n-1}\)

## Examples Using Arithmetic Sequence Explicit Formula

**Example 1:**** **Find the 15^{th} term of the arithmetic sequence -3, -1, 1, 3, ..... using the arithmetic sequence explicit formula.

**Solution:**

The first term of the given sequence is, a = -3

The common difference is, d = -1 - (-3) (or) 1 - (-1) (or) 3 - 1 = 2.

The 15^{th} term of the given sequence is calculated using,

\(a_n\) = a + (n - 1) d

\(a_{15}\) = -3 + (15 - 1) 2 = -3 + 14(2) = -3 + 28 = 25.

**Answer: **The 15^{th} term of the given sequence = 25.

**Example 2: **Find the common difference of the arithmetic sequence whose first term is 1/2 and whose 10^{th} term is 9.

**Solution:**

The first term is, a = 1/2.

Its 10^{th} term is a\(_{10}\) = 9.

Using the arithmetic sequence explicit formula,

\(a_n\) = a + (n - 1) d

Substituting n = 10, we get

\(a_{10}=a+(n-1)d\)

9 = (1/2) + (10 - 1) d

9 = (1/2) + 9d

Subtracting 1/2 from both sides,

17/2 = 9d

Dividing both sides by 9,

d = 17/18

**Answer: **The common difference is 17/18.

**Example 3: **Find the general term (or) n^{th} term of the arithmetic sequence -1/2, 2, 9/2, .....

**Solution:**

The first term of the given sequence is, a = -1/2.

The common difference is d = 2 - (-1/2) (or) 9/2 - 2 = 5/2.

We can find the general term (or) n^{th} term of an arithmetic sequence using the arithmetic sequence explicit formula.

\(a_n\) = a + (n - 1) d

\(a_n\) = -1/2 + (n - 1) (5/2)

\(a_n\) = - 1/2 + 5/2 n - 5/2

\(a_n\) = (5/2) n - 3

**Answer: **The n^{th} term of the given sequence = (5/2) n - 3.

## FAQs on Arithmetic Sequence Explicit Formula

### What Is Arithmetic Sequence Explicit Formula?

The arithmetic sequence explicit formula is a formula that is used to find the n^{th} term of an arithmetic sequence without computing any other terms before the n^{th} term. Using this formula, the n^{th} term of an arithmetic sequence whose first term is 'a' and common difference is 'd' is, a\(_n\) = a + (n - 1) d.

### How To Derive Arithmetic Sequence Explicit Formula?

We know that the differences between every two consecutive terms of an arithmetic sequence are all same. Thus, an arithmetic sequence is of the form a, a + d, a + 2d, .... If we observe here, the 1^{st }term is a = a + (1 - 1) d, the 2^{nd} term is a + d = a + (2 - 1) d, and the 3^{rd} term is a + 2d = a + (3 - 1) d. In the same way, the n^{th} term is, a\(_n\) = a + (n - 1) d.

### What Are the Applications of Arithmetic Sequence Explicit Formula?

Using the arithmetic sequence explicit formula, we can find any term of an arithmetic sequence just with the help of the first term and the common difference. For example, to find the 50^{th} term of an arithmetic sequence -7, -5, -3, ..., we just use the first term, a = -7 and the common difference, d = 2 and substitute in the formula a\(_n\) = a + (n - 1) d, then we get a\(_{50}\) = (-7) + (50 - 1) 2 = 91.

### What Is the 100^{th} term of the Arithmetic Sequence 1/4, 1/2, 3/4, ... Using Arithmetic Sequence Explicit Formula?

In the given sequence, a = 1/4 and the common difference is, d = 1/2 - 1/4 = 3/4 - 1/2 = ... = 1/4. Substituting n = 100 in the arithmetic sequence explicit formula, we get

a\(_n\) = a + (n - 1) d

a\(_{100}\) = (1/4) + (100 - 1) (1/4) = 25.