How to Find a Number of Terms in Arithmetic Sequence

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Equal intervals, called the common difference, separate the terms of an arithmetic sequence. The common difference can be either a positive or a negative number. For example, [2, 5, 8, 11...] is an arithmetic sequence with a common difference of 3, while [2, -1, -4, -7...] is an arithmetic sequence with a common difference of -3. In order to find a term in an arithmetic sequence, you need a starting point and the common difference. The general definition of the arithmetic sequence is a(n) = a(1) + (n - 1)d.

Things You'll Need

  • Calculator (optional)
  • Define the starting point. This is critical because all the terms you find have to be referenced to the same starting point. This will either be a part of the problem statement or you will have to choose it. Put the starting value in the arithmetic sequence equation.

    a(1) = 3

    a(n) = 3 + (n - 1)d

  • Define the common difference. Like the starting point, this will either be part of the problem or you will have to choose it. Put it into the equation.

    d = 5

    a(n) = 3 + (n - 1)5

  • Decide which terms you want to calculate. These will be the values for "n" in the equation. For this example, use n = [3, 7, 9, 11] for the third, seventh, ninth, and eleventh terms.

  • Calculate the value in the arithmetic sequence for each value of n.

    a(3) = 3 + (3 - 1)5 =13
    a(7) = 3 + (7 - 1)5 = 33
    a(9) = 3 + (9 - 1)5 = 43
    a(11) = 3 + (11 - 1)5 = 53

  • Check your work, if practical, by writing out the series to include all of your values for "n." This will not be practical for very high values of n but it can increase confidence when using lower values of n.

    a(n) = 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58

    The third, seventh, ninth, and eleventh terms in this sequence match the calculations from the previous step.

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