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

By Yang Kuang, Elleyne Kase

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:

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


    Using a system of two formulas to find the value of the variable.

    You can use substitution to solve one equation for a1:

    Solve one of the equations for one of the two variables.

    Plug this expression in for a1 in the other equation:

    PLug in the solved equation in the other one.

    Now simplify this equation:

    Simplify the new equation

  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

      Once you have found the value for one of the variables, solve the equation for the other.

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

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

      A formula without unknown variables.

  3. Find the term you’re looking for.

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

    Finding the value of a specific term in a geometric sequence.

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,

A geometric sequence.

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