The first term in an arithmetic sequence, denoted as a1, can be found using the formula: an = a1 + d(n - 1), where an is a known term in the sequence, d is the common difference, and n is the position of an in the sequence.
Understanding the Formula
The arithmetic sequence formula is:
an = a1 + d(n - 1)
- an: This is the nth term (a known term) in the sequence.
- a1: This represents the first term of the sequence, which is what we want to find.
- d: This is the common difference between consecutive terms in the sequence.
- n: This is the position of the term an in the sequence.
How to find a1
To find a1 we need to rearrange the formula above to isolate a1:
a1 = an - d(n - 1)
Steps to find a1
Here's a breakdown of how to find the first term:
- Identify a Known Term (an): Find any term in the sequence that you know the value of (e.g., the 5th term is 20).
- Determine the Common Difference (d): Calculate the difference between any two consecutive terms in the sequence. If the sequence is 2, 4, 6, 8..., then d = 2.
- Identify the Position (n) of the Known Term: Determine the position of the term an that you identified in step 1. If you know the 5th term, then n = 5.
- Substitute and Solve: Plug the values of an, d, and n into the rearranged formula: a1 = an - d(n - 1) and solve for a1.
Example
Let's say you know the 7th term of an arithmetic sequence is 31, and the common difference is 4. Find the first term.
- an = 31
- d = 4
- n = 7
Now, substitute these values into the formula:
a1 = 31 - 4(7 - 1)
a1 = 31 - 4(6)
a1 = 31 - 24
a1 = 7
Therefore, the first term of the arithmetic sequence is 7.
Practical Insights
- If you only know two terms that are not consecutive, you can still find the common difference. Divide the difference between the two terms by the difference in their positions. For example, if the 3rd term is 10 and the 6th term is 19, then d = (19-10)/(6-3) = 9/3 = 3.
- Understanding the formula and being able to rearrange it is crucial for solving various arithmetic sequence problems.
