Arithmetic Sequence Explicit Formula - Derivation of Arithmetic

The arithmetic sequence explicit formula is a valuable tool for finding any term in a given arithmetic sequence. An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This common difference is denoted as 'd.'

Let's take an example: consider the arithmetic sequence 2, 5, 8, 11, .... In this sequence:

The first term (a) is 2.

The common difference (d) is 3, which is calculated as 5 - 2 = 3.

The arithmetic sequence explicit formula is given as:

an = a + (n - 1)d

This formula allows us to find the nth term (an) of the sequence without needing to know the previous term. In this case, with the values provided:

an = 2 + (n - 1)3

Simplifying further:

an = 3n - 1

So, the explicit formula for this arithmetic sequence is an = 3n - 1. This formula is a valuable tool for calculating specific terms within the sequence.

Let's explore the arithmetic sequence explicit formula in more detail, including its derivation and examples, to understand its practical applications better.

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What Is Arithmetic Sequence Explicit Formula?

The arithmetic sequence explicit formula serves as a powerful tool for determining any term within an arithmetic sequence (a1, a2, a3, ..., an, ...). It relies on two key parameters: the first term (a) and the common difference (d) that characterize the sequence. The formula itself is expressed as an = a + (n - 1)d, offering a straightforward means of calculating the nth term of an arithmetic sequence.

Derivation of Arithmetic Sequence Explicit Formula

The arithmetic sequence explicit formula finds its roots in the terms of the arithmetic sequence, offering a convenient method for determining any specific term within the sequence. In an arithmetic sequence represented as a1, a2, a3, ..., an, the initial term is denoted as 'a,' with a1 being equivalent to 'a.' Additionally, the common difference is symbolized as 'd,' and its formula is derived as follows: d = a2 - a1 = a3 - a2 = an - an-1 . The nth term of the arithmetic sequence corresponds to the explicit formula, which can be expressed as follows:

Explicit Formula: an = a + (n - 1)d

Here, the parameters are defined as follows:

an: The nth term of the arithmetic sequence.

a: The first term of the arithmetic sequence.

d: The common difference, which signifies the difference between each term and its preceding term (i.e., d = an - an-1) .

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Example of Arithmetic Sequence Explicit Formula

An illustrative example of an arithmetic sequence is provided by the sequence -3, -6, -9, -12, ..., in which the common difference, often denoted as 'd,' is consistently -3. The first term of this arithmetic sequence, which can be represented as a1 or simply 'a,' is equal to -3. Utilizing the arithmetic sequence explicit formula for the nth term, we arrive at the following calculation: an = a + (n - 1)d = -3 + (n - 1)(-3) = -3n + 3 - 3 = -3n. In this case, the explicit formula for the nth term of the sequence is succinctly expressed as an = -3n.

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Examples Using Arithmetic Sequence Explicit Formula

Example 1: Let's calculate the 15th term of the arithmetic sequence -3, -1, 1, 3, ... using the arithmetic sequence explicit formula.

Solution:

The first term of the sequence is a = -3.

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

To find the 15th term (a15), we use the formula:

an = a + (n - 1)d

a15 = -3 + (15 - 1) × 2 = -3 + 14 × 2 = -3 + 28 = 25.

Answer: The 15th term of the sequence is 25.

Example 2: Find the common difference of an arithmetic sequence with a first term of 1/2 and a 10th term of 9.

Solution:

The first term is a = 1/2.

The 10th term (a10) is 9.

Using the arithmetic sequence explicit formula:

an = a + (n - 1)d

For n = 10:

a10 = (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 nth term) of the arithmetic sequence -1/2, 2, 9/2, ...

Solution:

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

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

To find the general term (an) of an arithmetic sequence, we use the arithmetic sequence explicit formula:

an = a + (n - 1)d

an = -1/2 + (n - 1)(5/2)

an = -1/2 + (5/2)n - 5/2

an = (5/2)n - 3

Answer: The nth term of the sequence is (5/2)n - 3.