Finding the common ratio in a geometric progression (GP) is straightforward. A geometric progression is a sequence where each term is found by multiplying the previous term by a constant value, known as the common ratio.
Calculating the Common Ratio
To find the common ratio (often denoted as 'r'), simply divide any term by the preceding term.
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Formula:
r = aₙ / aₙ₋₁whereaₙis the nth term andaₙ₋₁is the (n-1)th term. -
Method: Choose any two consecutive terms in the GP. Divide the later term by the earlier term. This quotient is the common ratio. Repeat this process with other pairs of consecutive terms to verify consistency.
Examples
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Example 1: Consider the GP: 2, 6, 18, 54...
r = 6 / 2 = 3r = 18 / 6 = 3r = 54 / 18 = 3
The common ratio is 3.
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Example 2: Consider the GP: 100, 50, 25, 12.5...
r = 50 / 100 = 0.5r = 25 / 50 = 0.5r = 12.5 / 25 = 0.5
The common ratio is 0.5.
Finding the Common Ratio with Limited Information
If you don't have consecutive terms, but know other information, different methods may be needed. For instance, if you know the first and last terms, along with the number of terms, you can use the formula for the nth term of a GP to solve for 'r'. However, this is beyond the scope of a simple explanation of finding the common ratio using consecutive terms.
Key Considerations
- The common ratio can be positive or negative. A negative common ratio indicates that the terms alternate in sign.
- The common ratio can be greater than 1 (terms increase), equal to 1 (all terms are equal), or between 0 and 1 (terms decrease).
