# Arithmetic Sequence Explicit Formula

Arithmetic sequence explicit formula is useful to find any terms of the given arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the differences between any two consecutive numbers are the same. For an arithmetic sequence 2, 5, 8, 11, .... the first term is a = 2, and the common difference is d = 5 - 2 = 3. The **arithmetic sequence explicit formula** for this series is a_{n} = a + (n - 1)d, or a_{n} = 2 + (n - 1)3 or a_{n} = 3n - 1.

Here the arithmetic sequence explicit formula (a_{n} = 3n - 1) is useful to find any terms of the series and can be calculated without knowing the previous term. Let us learn the arithmetic sequence explicit formula, and its derivation with the help of examples, FAQs.

## 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 a_{n} = a + (n - 1)d.

### Derivation of Arithmetic Sequence Explicit Formula

The arithmetic sequence explicit formula is derived from the terms of the arithmetic sequence. It helps to easily find any term of the arithmetic sequence. The arithmetic sequence is a_{1}, a_{2}, a_{3}, ..., a_{n}. Here the first term is referred as 'a' and we have a = a_{1} and the common difference is denoted as 'd'. The formula for the common difference is d = a_{2} - a_{1} = a_{3} - a_{2} = a_{n} - a_{n - 1}. The n^{th } term of the arithmetic sequence represents the explicit formula of the arithmetic sequence.

### Explicit Formula: 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 \(=a_n-a_{n-1}\)

### Example of Arithmetic Sequence Explicit Formula

An example of an arithmetic sequence is -3, -6, -9, -12, ... , for which the difference between any term and its previous term is -3. This difference is usually termed as the common difference and is denoted by d. The first term of an arithmetic sequence is represented by a_{1} or a = -3. Applying the formula for the n^{th} term, the arithmetic series explicit formula is a_{n} = a + (n - 1)d = -3 + (n - 1)(-3) = -3n + 3 - 3 = -3n, or a_{n} = -3n.

## 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, we have the following 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.

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