How do you find the geometric progression ratio?

To find the geometric progression ratio (often called the "common ratio"), you need to identify a geometric sequence and understand how each term relates to the previous one. The common ratio is the value that each term is multiplied by to get the next term.

Understanding Geometric Progressions

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

• Formula: The general formula for the nth term in a geometric progression is: an = ar^(n-1), where:
  • an is the nth term
  • a is the first term
  • r is the common ratio
  • n is the position of the term

Methods for Finding the Common Ratio

Here are two primary methods to calculate the common ratio (r):

1. Dividing Consecutive Terms:

   • If you have any two consecutive terms in the sequence, divide the later term by the earlier term. This ratio will be the same no matter which pair of consecutive terms you choose.
   • Example: In the sequence 2, 6, 18, 54...,
     • r = 6 / 2 = 3
     • r = 18 / 6 = 3
     • r = 54 / 18 = 3
     • Therefore, the common ratio is 3.

2. Using the Formula with Known Terms:

   • If you know specific terms, like the 3rd term (a3) and the 1st term (a), you can utilize the formula an = ar^(n-1)

     • Example: Suppose the first term (a) is 5 and the third term (a3) is 45.
     1. Use the formula for the third term: a3 = ar^(3-1)
     2. Substitute the known values: 45 = 5 * r^2
     3. Solve for r:
        • Divide both sides by 5: 9 = r^2
        • Take the square root of both sides: r = ±3
        • Therefore, the common ratio is either 3 or -3. This indicates two possible sequences: one increasing (5, 15, 45...) and one alternating (5, -15, 45...).

Practical Insights

• Non-Zero Ratio: The common ratio must not be zero. If it were zero, all terms after the first would be zero, which is not a geometric progression.
• Positive or Negative: The common ratio can be either positive or negative. A positive ratio creates a sequence with all positive terms (or all negative terms), while a negative ratio alternates the sign of the terms.
• Fractional Ratios: The common ratio can also be a fraction, which results in a decreasing sequence (in absolute value) or an alternating sequence decreasing in absolute value.

Summary Table

Method	Description	Example
Dividing Consecutive Terms	Divide any term by the preceding term in the sequence.	r = a₂ / a₁ or r = a₃/a₂
Using the Formula with Known Terms	Utilize the formula an = ar^(n-1) and solve for r using known terms and their positions.	45 = 5r^2; r=±3

In short, you find the geometric progression ratio by consistently dividing any term by its preceding term.