How to Identify a Term in a Geometric Sequence When You Know Two Nonconsecutive Terms

If your pre-calculus teacher gives you any two nonconsecutive terms of a geometric sequence, you can find the general formula of the sequence as well as any specified term. For example, if the 5th term of a geometric sequence is 64 and the 10th term is 2, you can find the 15th term. Just follow these steps:

  1. Determine the value of r.

    You can use the geometric formula to create a system of two formulas to find r:

    image0.png

    or

    image1.png

    You can use substitution to solve one equation for a1:

    image2.png

    Plug this expression in for a1 in the other equation:

    image3.png

    Now simplify this equation:

    image4.png
  2. Find the specific formula for the given sequence.

    1. a.Plug r into one of the equations to find a1.

      This step gives you

      image5.png
    2. b.Plug a1 and r into the formula.

      Now that you know a1 and r, you can write the formula:

      image6.png
  3. Find the term you're looking for.

    In this case, you want to find the 15th term (n = 15):

    image7.png

Here's another example: The annual depreciation of a car's value is approximately 30 percent. Thus, every year, the car is actually worth 70 percent of its value from the year before. If a1 represents the value of a car when it was new and n represents the number of years that have passed,

image8.png

Notice that this sequence starts at 0, which is okay as long as the information says that it starts at 0.

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