How can I determine the first term of an arithmetic sequence if I know two of its terms?

by

Finding the First Term of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference (
d
). If you are given two terms from this sequence, you can easily find the first term.

Step-by-step Method

  1. Identify the Given Terms: Let’s say you have two terms:
    am and
    a
    , where
    m and
    n
    are their positions in the sequence.
  2. Calculate the Common Difference: The common difference can be calculated using the formula:

    d = (a - a) / (n - m)
  3. Determine the First Term: Once you have the common difference, you can express the first term (
    a1
    ) of the sequence as follows:

    a1 = a - (m - 1) * d

Example

For instance, if you know the 3rd term (
a3 = 10
) and the 5th term (
a5 = 18
) of an arithmetic sequence, you can find the first term as follows:

  1. Calculate the common difference:

    d = (18 - 10) / (5 - 3) = 4
  2. Now, substitute back to find the first term:

    a1 = 10 - (3 - 1) * 4 = 10 - 8 = 2

Thus, in this example, the first term of the arithmetic sequence is 2.

Leave a Comment