How to Find a Number of Terms in an Arithmetic Sequence

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Finding the number of terms in an arithmetic sequence might sound like a complex task, but it’s actually pretty straightforward. All you need to do is plug the given values into the formula t n = a + (n - 1) d and solve for n , which is the number of terms. Note that t n is the last number in the sequence, a is the first term in the sequence, and d is the common difference. [1]

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    Typically, to solve a problem like this, you’ll be given the first 3 or more terms as well as the last term. [2]
    • For example, you may have the following sequence: 107, 101, 95…-61. In this case, the first term is 107, the second term is 101, and the last term is -61. You need all of this information to solve the problem.
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    In the example sequence, the first term is 107 and the second term is 101. So, subtract 107 from 101, which is -6. Therefore, the common difference is -6. [3]
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    Plug in the last term ( t n ), the first term ( a ), and the common difference ( d ). [4] Work through the equation until you’ve solved for n . [5]
    • For example, start by writing: -61 = 107 + (n - 1) -6. Subtract 107 from both sides so you’re left with -168 = (n - 1) -6. Then, divide both sides by -6 to get 28 = n - 1. Finish by adding 1 to both sides so that n = 29. [6]
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  • Question
    Can I use the formula (A - L/d) + 1 for finding n?
    Donagan
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    As explained above, n = [(L - A) / d] + 1.
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  • Question
    If the difference is not given, how do I find the difference?
    Donagan
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    Subtract any term from the one that follows it.
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  • Question
    If the first term and last term of an arithmetic progression are 5 and 89, how do I find the number of terms?
    Donagan
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    You can't without knowing the difference between consecutive terms.
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      References

      1. ↑ https://thirdspacelearning.com/gcse-maths/algebra/arithmetic-sequence/
      2. ↑ https://youtu.be/hGiuMNIR7Gk?t=13
      3. ↑ https://www.alamo.edu/contentassets/afe30946fa58450c89840c1173f3b9d0/sequences/math1314-arithmetic-sequences.pdf
      4. ↑ https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
      5. ↑ https://youtu.be/hGiuMNIR7Gk?t=59
      6. ↑ https://thirdspacelearning.com/gcse-maths/algebra/arithmetic-sequence/

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