How do you work out the number of terms in an arithmetic sequence?

To determine the number of terms in an arithmetic sequence, you need to know the first term, the last term (if it exists), and the common difference. The video reference mentions that arithmetic sequences don't always have a last term; sometimes they go on forever (0:30-4:05). However, if there is a last term, we can use the following to determine the number of terms.

Here's how to calculate the number of terms in a finite arithmetic sequence:

1. Identify the first term (a₁), the last term (a_n), and the common difference (d). The common difference is the value you add to each term to get the next term in the sequence.

2. Use the arithmetic sequence formula:

   a_n = a₁ + (n - 1)d

   where:

   • a_n is the last term of the sequence.
   • a₁ is the first term of the sequence.
   • n is the number of terms in the sequence (what we're trying to find).
   • d is the common difference between terms.

3. Rearrange the formula to solve for n:

   n = (a_n - a₁) / d + 1

4. Substitute the known values into the formula and calculate n.

Example:

Let's say we have an arithmetic sequence: 2, 5, 8, ..., 29.

• a₁ = 2 (the first term)
• a_n = 29 (the last term)
• d = 3 (the common difference, since 5 - 2 = 3 and 8 - 5 = 3)

Now, we plug these values into the formula:

n = (29 - 2) / 3 + 1

n = 27 / 3 + 1

n = 9 + 1

n = 10

Therefore, there are 10 terms in the arithmetic sequence.