Geometric Sequence Basic - Algebra-1

1. Fundamental Concepts

Geometric Sequence Definition：A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio (denoted by r).
Common Ratio (r): It is calculated by dividing any term in the sequence by its preceding term, i.e., \(r=\frac{a_{n}}{a_{n-1}}\) for \(n\geq2\), where \(a_n\) is the n-th term and \(a_{n-1}\) is the \((n-1)\)-th term.

Identification of Geometric Sequence: A sequence is geometric if the ratio between every pair of consecutive terms is constant. If the ratio varies, it is not a geometric sequence.
Role of Common Ratio: The value of r determines the behavior of the sequence:
- If |r| > 1, the terms grow in magnitude.
- If |r| < 1 (and \(r\neq0\)), the terms shrink in magnitude.
- If \(r = 1\), all terms are equal.
- If \(r=-1\), the terms alternate between positive and negative with the same magnitude.

Question: Is the sequence \(2, 6, 18, 54, ...\) geometric?
Solution: Calculate the ratios between consecutive terms:
- \(\frac{6}{2}=3\)
- \(\frac{18}{6}=3\)
- \(\frac{54}{18}=3\)
Conclusion: The ratio is constant (\(r = 3\)), so it is a geometric sequence.

Question: Determine if \(8, -4, 2, -1, ...\) is a geometric sequence.
Solution: Find the ratios:
- \(\frac{-4}{8}=-\frac{1}{2}\)
- \(\frac{2}{-4}=-\frac{1}{2}\)
- \(\frac{-1}{2}=-\frac{1}{2}\)
Conclusion: The common ratio \(r=-\frac{1}{2}\) is constant, so it is a geometric sequence.

Question: Is the sequence \(3, 6, 12, 24, 47, ...\) geometric?
Solution: Check the ratios:
- \(\frac{6}{3}=2\)
- \(\frac{12}{6}=2\)
- \(\frac{24}{12}=2\)
- \(\frac{47}{24}\approx1.958\) (not equal to 2)
Conclusion: The ratio is not constant, so it is not a geometric sequence.

Calculate Consecutive Ratios: For a given sequence \(a_1, a_2, a_3, ..., a_n\), compute \(\frac{a_2}{a_1}, \frac{a_3}{a_2}, ..., \frac{a_n}{a_{n-1}}\).
Check Consistency: If all these ratios are equal, the sequence is geometric; otherwise, it is not.
Note Special Cases: If a sequence has a term equal to 0 (except possibly the first term, but even then, the next term would be 0, leading to undefined ratios when dividing by 0), it cannot be geometric.