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:

   

   or

   

   You can use substitution to solve one equation for a₁:

   

   Plug this expression in for a₁ in the other equation:

   

   Now simplify this 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

      

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

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

      

3. Find the term you're looking for.

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

   

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,

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