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:
Determine the value of r.
You can use the geometric formula to create a system of two formulas to find r:
You can use substitution to solve one equation for a1:
Plug this expression in for a1 in the other equation:
Now simplify this equation:
Find the specific formula for the given sequence.
a.Plug r into one of the equations to find a1.
This step gives you
b.Plug a1 and r into the formula.
Now that you know a1 and r, you can write the formula:
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.