The sequence -2, 6, -18, 54,... follows a geometric progression where each term is multiplied by a fixed number to reach the next term. Identifying this fixed multiplier, known as the common ratio, allows you to predict any term in the sequence.
Understanding the pattern in this alternating sign sequence helps solve problems in finance, physics, and computer science where exponential growth or decay appears.
| Term Position | Term Value | Calculation from Previous Term | Multiplier |
|---|---|---|---|
| 1 | -2 | Initial term | - |
| 2 | 6 | -2 × (-3) | -3 |
| 3 | -18 | 6 × (-3) | -3 |
| 4 | 54 | -18 × (-3) | -3 |
| 5 | -162 | 54 × (-3) | -3 |
Recognizing Geometric Patterns
Each term in this sequence is created by multiplying the previous term by the same number. This consistent structure defines a geometric sequence and makes it distinct from arithmetic patterns where addition is used.
Spotting the geometric form early allows you to write a general formula and compute terms far down the line without listing every intermediate value.
Calculating the Common Ratio
Divide any term by the term before it to find the common ratio. Using the second term divided by the first term, 6 ÷ (-2) equals -3. Verifying with later terms, such as -18 ÷ 6, also yields -3, confirming consistency across the sequence.
The negative ratio indicates that the terms alternate in sign, switching from negative to positive and back with each step.
Formula and General Term
The general term of a geometric sequence can be written as a_n = a_1 × r^(n-1), where a_1 is the first term and r is the common ratio. For this sequence, a_1 is -2 and r is -3, so the formula becomes a_n = -2 × (-3)^(n-1).
Using this formula, you can directly calculate the 10th or 100th term without manually stepping through every previous term.
Behavior and Magnitude Growth
Although the ratio has an absolute value greater than 1, causing the magnitude of terms to grow quickly, the alternating sign creates a wave-like pattern between negative and positive values. This mix of rapid growth and sign switching appears in contexts such as certain wave equations and alternating current scenarios.
Understanding both the size and sign behavior helps interpret real-world systems modeled by this sequence.
Key Takeaways
- The common ratio of the sequence is consistently -3.
- Each term is obtained by multiplying the previous term by -3.
- The general term formula is a_n = -2 × (-3)^(n-1).
- The sequence alternates in sign while its magnitude grows exponentially.
- Verification across multiple term pairs ensures the ratio is accurate.
FAQ
Reader questions
How do I verify the common ratio for this sequence?
Divide any term by the term directly before it. For example, 54 divided by -18 equals -3, and -18 divided by 6 also equals -3, confirming the ratio is consistently -3.
Can the common ratio be used to find any term in the sequence?
Yes, using the formula a_n = -2 × (-3)^(n-1), where n is the term number, lets you compute any term directly without building the entire list.
What would the next two terms after 54 be?
Multiply 54 by -3 to get -162, then multiply -162 by -3 to get 486, so the next two terms are -162 and 486.
Does the sequence ever reach zero?
No, because you start with -2 and repeatedly multiply by -3, the terms never become zero and the sequence diverges in magnitude.