How to Find the First Term in an Arithmetic Sequence

Understanding how to identify and calculate the first term of an arithmetic sequence is crucial for many mathematical applications. An arithmetic sequence is defined as a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. Below, we will explore the steps to find the first term, a1, in an arithmetic sequence when given specific information about the sequence.

Understanding the Formula for an Arithmetic Sequence

The formula for the n-th term (a_n) in an arithmetic sequence is:

a_n a1 (n - 1) middot; d

Where:

a1 represents the first term of the sequence. n represents the position of the term in the sequence. d is the common difference, representing the constant difference between consecutive terms.

Deriving the Formula to Find the First Term

To find the first term a1, we can rearrange the formula:

a1 a_n - (n - 1) middot; d

This formula allows you to determine the first term if you know the n-th term, the position of the term, and the common difference.

Steps to Find the First Term in an Arithmetic Sequence

Identify the Known Information: You need to know three key pieces of information: the n-th term, the position of that term in the sequence, and the common difference. Determine the Values: Assign the given values to the appropriate variables in the formula. For example, if you know the 5th term a5 20 and the common difference d 3. Plug in the Values: Substitute the values into the formula to solve for the first term. Using the example provided: Calculate: Perform the arithmetic calculations to determine the first term.

Example:
Given that the 5th term a5 20 and the common difference d 3:

n 5
a1 20 - (5 - 1) middot; 3
a1 20 - 4 middot; 3
a1 20 - 12 8

Thus, the first term a1 is 8.

Identifying the Pattern

Often, by observing the second term and subsequent terms in the sequence, you can easily identify the pattern. If the sequence follows a consistent, predictable pattern, it can help you to confirm and calculate the first term accurately.

Understanding Geometric vs. Arithmetic Sequences

It's worth noting that while arithmetic sequences involve addition, geometric sequences involve multiplication. In a geometric sequence, you need both the first term and the multiplier (common ratio) to find the first term. However, in an arithmetic sequence, as long as you know the common difference and a term in the sequence, you can work backwards to find the first term.

Both types of sequences have their unique properties, but the method to find the first term in an arithmetic sequence as described above is commonly applicable and straightforward.