The first term of an arithmetic sequence is often designated as a1 and is a foundational element for understanding the entire sequence. The reference video on arithmetic sequences ([Arithmetic Sequences and Arithmetic Series - Basic Introduction - YouTube]) highlights the importance of knowing this term, as it is used to define the formula of an arithmetic sequence. Here’s how to approach finding the first term:
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between any two successive members is a constant. This constant difference is called the 'common difference' and is usually denoted by d.
The General Formula
The general formula, also known as the explicit formula, for an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term in the sequence.
- a1 is the first term in the sequence, which we want to find.
- n is the position of the term in the sequence.
- d is the common difference.
Finding the First Term (a1)
The reference video states that to use the general formula, you need the first term and the common difference. Therefore, if you know any term of the sequence (an), its position in the sequence (n), and the common difference (d), you can rearrange the general formula to solve for a1.
Rearranging the Formula:
To find a1, you can rearrange the formula as follows:
a1 = an - (n - 1)d
Steps to Find the First Term:
- Identify a term and its position: Identify any term in the sequence, an, and its corresponding position n. For example, if the third term is 7, then a3 = 7 and n = 3.
- Find the common difference (d): Calculate the common difference by subtracting any term from its subsequent term (d = an+1 - an).
- Apply the rearranged formula: Substitute the known values into the formula a1 = an - (n - 1)d to find the first term.
Example:
Suppose we have an arithmetic sequence where the third term (a3) is 7 and the common difference (d) is 2.
- We know that a3 = 7 and n = 3.
- We are given that d = 2.
- Using the formula a1 = an - (n - 1)d, we can calculate the first term as:
a1 = 7 - (3 - 1) 2
a1 = 7 - 2 2
a1 = 7 - 4
a1 = 3
Therefore, the first term of this sequence is 3.
Summary:
| Component | Description |
|---|---|
| a1 | The first term of the arithmetic sequence, which is what we are trying to find. |
| an | Any term in the sequence other than the first term. |
| n | The position of the chosen term within the sequence. |
| d | The constant difference between any two successive terms (the common difference). |
| Formula | a1 = an - (n - 1)d |
By understanding the relationship between these components, finding the first term of any arithmetic sequence becomes straightforward. It’s all about applying the correct formula after determining a known term, its position, and the common difference.
