Finding the common ratio is straightforward for a geometric sequence. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a constant value; this constant is the common ratio.
Calculating the Common Ratio
To find the common ratio (often denoted as r), simply divide any term by the term before it.
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Formula:
r = aₙ / aₙ₋₁whereaₙis the nth term andaₙ₋₁is the (n-1)th term. -
Example: Consider the geometric sequence: 2, 6, 18, 54...
r = 6 / 2 = 3r = 18 / 6 = 3r = 54 / 18 = 3
The common ratio is 3.
Finding the Common Ratio with Limited Information
Sometimes, you might not have consecutive terms. If you know the first term (a₁) and another term (aₙ) along with its position (n), you can use the general formula for the nth term of a geometric sequence:
- Formula:
aₙ = a₁ * rⁿ⁻¹
You can solve this equation for r:
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Solving for r:
r = (aₙ / a₁) ^ (1/(n-1)) -
Example: If a₁ = 2, a₅ = 32, and n = 5, then:
r = (32 / 2) ^ (1/(5-1)) = 16 ^ (1/4) = 2
The common ratio is 2.
As stated in various sources like Cuemath, the common ratio is the constant multiplier between consecutive terms in a geometric sequence. Several YouTube videos (Find the Common Ratio of a Geometric Sequence - YouTube, Find the Common Ratio of a Geometric Sequence - YouTube) also demonstrate this method. Remember that the sequence must be geometric for this method to apply.
