How to find the first term in an arithmetic sequence?

To find the first term of an arithmetic sequence, you often need to work backward from a known term using the common difference. Here's how you can approach it, drawing on the concepts from the provided reference:

Understanding Arithmetic Sequences

An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the "common difference" (often denoted as 'd').

Using a Known Term and the Common Difference

• The General Formula: The formula for the nth term of an arithmetic sequence is:
  an = a1 + (n - 1)d
  Where:

  • an is the nth term
  • a1 is the first term (what we want to find)
  • n is the term number
  • d is the common difference

• Rearranging to Find the First Term (a1): If you know a term ( an), its position ( n), and the common difference (d), you can rearrange the formula to solve for a1:
  a1 = an - (n - 1)d

Example based on reference:

The provided video excerpt shows a problem where the sixth term is known and we are aiming to get the tenth term.
The excerpt notes: "If we're at the sixth sixth term and we're trying to get to the tenth term, we have to add d four times."

• This translates to a10 = a6 + 4d.
• However, if we knew a10, we could rearrange it to a6 = a10 - 4d.

And working backwards to find a1 would require repeating this concept.

How to find a1 if we only know a later term:

1. Identify known values: Determine the value of a known term (an), its position in the sequence (n), and the common difference (d).
2. Apply the formula: Use the formula a1 = an - (n - 1)d.
3. Calculate a1: Substitute the known values into the formula and calculate the value of a1.

Example:

Let's say we know the 6th term of an arithmetic sequence is 15 and the common difference is 2.

1. Known Values:

   • a6 = 15
   • n = 6
   • d = 2

2. Apply Formula:
   a1 = a6 - (6 - 1)d
a1 = 15 - (5) * 2

3. Calculate a1:
   a1 = 15 - 10
a1 = 5

Therefore, the first term of this sequence is 5.

Key Takeaways

• To find the first term (a1), you need at least one known term, its position, and the common difference (d).
• The formula a1 = an - (n - 1)d is the key to solving these types of problems.