Common Ratio - Algebra-1

1. Fundamental Concepts

Definition of Common Ratio: In a geometric sequence, the ratio of each term (starting from the second term) to its preceding term is a fixed constant, which is called the common ratio, usually denoted by the letter r.
Mathematical Expression of Common Ratio: For a geometric sequence (where is the first term and is the n-th term), the common ratio (for ).

Uniqueness of Common Ratio: In a given geometric sequence, the common ratio is unique, meaning the ratio between all adjacent terms is equal, and there are no multiple different common ratios.
Range of Common Ratio: The common ratio r can be positive, negative, or a number with an absolute value greater than 1, equal to 1, or less than 1 (but it cannot be 0; because if , all subsequent terms except the first term will be 0, making it impossible to calculate a valid common ratio).
Influence of Common Ratio on the Sequence:
- When , the absolute values of the terms in the sequence will increase as the number of terms increases.
- When (and ), the absolute values of the terms in the sequence will decrease as the number of terms increases.
- When , every term in the sequence is equal, forming a constant sequence.
- When , the terms in the sequence alternate between positive and negative, with equal absolute values.

Question: Find the common ratio r in the geometric sequence .
Solution: Calculate according to the definition of common ratio:
- Verification: ,
Conclusion: The common ratio of the sequence is .

Question: Given that the first three terms of a geometric sequence are , find the common ratio r.
Solution:
- Verification:
Conclusion: The common ratio of the sequence is .

Finding the common ratio when adjacent terms are known: Directly divide the latter term by the former term, that is, (for ).
Verifying the correctness of the common ratio: After calculating the common ratio, substitute it into other adjacent terms in the sequence to verify that the ratio of all adjacent terms is equal to the calculated common ratio, so as to avoid errors due to miscalculations.