The common ratio in a geometric sequence is the constant value by which you multiply (or divide) each term to get the next term. In simpler terms, it's the consistent multiplier that defines the pattern of the sequence.
Understanding the Common Ratio
A geometric sequence progresses by either multiplying or dividing by the same number at each step. This consistent factor is what we call the common ratio.
- Multiplication: When the sequence's terms are increasing, each term is multiplied by the common ratio. For example, in the sequence 2, 6, 18, 54..., the common ratio is 3 (because 2 x 3 = 6, 6 x 3 = 18, and so on).
- Division: When the sequence's terms are decreasing, each term is, in essence, divided by the inverse of the common ratio (or multiplied by a fraction). For example, in the sequence 8, 4, 2, 1..., the common ratio is 1/2 (because 8 x 1/2 = 4, 4 x 1/2 = 2, and so on). You can also think of this as dividing by 2.
How to Find the Common Ratio
The reference states that the common ratio is found by dividing any two consecutive terms in the sequence.
- Calculation: You can calculate the common ratio (often represented as 'r') by dividing any term by its preceding term. So, if you have terms an and an-1, the common ratio, r, is:
r = an / an-1 - Consistency: It does not matter which consecutive pair of terms you choose, the common ratio should remain constant throughout the sequence.
Examples:
| Sequence | Calculation | Common Ratio |
|---|---|---|
| 3, 6, 12, 24... | 6/3 = 2, 12/6 = 2 | 2 |
| 100, 20, 4, 0.8... | 20/100 = 0.2, 4/20=0.2 | 0.2 |
| 1, -3, 9, -27... | -3/1 = -3, 9/-3 = -3 | -3 |
Key Takeaways
- The common ratio is a constant multiplier or divisor in a geometric sequence.
- It is found by dividing any term by its preceding term.
- The common ratio must be consistent throughout the entire sequence.
