To find the common ratio (r) in a geometric sequence of fractions, simply divide any term by its preceding term. This works because the common ratio is the constant value you multiply each term by to get the next term.
Understanding Geometric Sequences
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Finding the Common Ratio: A Step-by-Step Guide
-
Identify the Sequence: Make sure you have a geometric sequence. Check if the ratio between consecutive terms is constant.
-
Choose Two Consecutive Terms: Select any term in the sequence and the term immediately before it.
-
Divide: Divide the later term by the earlier term. This quotient is your common ratio (r).
-
Verify (Optional): To confirm, repeat step 3 with another pair of consecutive terms. The result should be the same common ratio.
Examples
-
Example 1: Consider the sequence: 1/2, 1/4, 1/8, 1/16...
- Divide the second term by the first: (1/4) / (1/2) = 1/2.
- Divide the third term by the second: (1/8) / (1/4) = 1/2.
- The common ratio (r) is 1/2.
-
Example 2: Consider the sequence: 5/12, -1/6, 1/15, -2/75...
- Divide the second term by the first: (-1/6) / (5/12) = -2/5
- Divide the third term by the second: (1/15) / (-1/6) = -2/5
- The common ratio (r) is -2/5.
-
Example 3 (from provided reference): Find the common ratio r for the sequence where the terms are 81/25 and 27/5.
- r = (27/5) รท (81/25) = (27/5) * (25/81) = 5/3
This demonstrates the method described in one of the provided references: "The more universal technique is to take the ratio of one term divided by its predecessor to see what the common ratio is."
Using the Common Ratio
Once you've found the common ratio, you can use it to:
- Find future terms: Multiply the last known term by the common ratio repeatedly.
- Find past terms: Divide the first known term by the common ratio repeatedly.
- Calculate the sum of the sequence: Use the formula for the sum of a geometric series. (This formula involves the common ratio.)
