Geometric Sequence (Constant Ratio)
Core Pattern: In a geometric sequence, each term is multiplied by a constant ratio $r$. When adjacent differences grow exponentially rather than staying steady, test division instead of subtraction.
Mathematical Formulation & Recurrence
\(x_n = x_1 \cdot r^{n-1} \quad \iff \quad x_n = x_{n-1} \cdot r \quad \left(r = \frac{x_n}{x_{n-1}}\right)\)
Method of Differences & Calculation Technique
Difference vs. Quotient Method: If you calculate differences \(\Delta_k = x_{k+1} - x_k\) on a geometric sequence (e.g. 3, 6, 12, 24 \(\to\) \(\Delta = 3, 6, 12\)), you notice the differences themselves form a geometric sequence! This explosive exponential gap proves the operation is multiplication. Switch from subtracting to dividing adjacent terms: \(r = x_{k+1} / x_k\). Once \(r\) is verified, find the next term: \(x_{\text{next}} = x_{\text{last}} \times r\).
Visual Sequence & Difference Mapping
How to Solve This Series in 3 Steps:
- 1 Notice rapid growth or decay: If numbers grow by doubling, tripling, or halve each step, test a ratio.
- 2 Compute the ratio r: Divide second term by first term: r = x₂ / x₁.
- 3 Confirm constant ratio: Ensure x₃ / x₂ = r.
- 4 Calculate next term: Multiply the last term by r (or divide if decaying).
Step-by-Step Worked Examples:
2, 6, 18, 54, 162, ?
Example 1
1. Quotients: 6/2=3, 18/6=3, 54/18=3
2. Common ratio r = 3
3. Next term: 162 × 3 = 486
Answer: 486
128, 64, 32, 16, 8, ?
Example 2
1. Quotients: 64/128=0.5 (or ÷2)
2. Decaying ratio r = 1/2
3. Next term: 8 / 2 = 4
Answer: 4
5, -15, 45, -135, ?
Example 3
1. Quotients: -15/5 = -3, 45/-15 = -3
2. Negative ratio r = -3
3. Next term: -135 × (-3) = 405
Answer: 405
4, 20, 100, 500, ?
Example 4
1. Quotients: 20/4 = 5, 100/20 = 5
2. Common ratio r = 5
3. Next term: 500 × 5 = 2500
Answer: 2500
Interactive Comprehension Test
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3, 12, 48, 192, ?
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