I determined the missing terms in geometric sequences by focusing on the consistent multiplicative relationship between consecutive terms. Here's a breakdown of the process:
Identifying the Common Ratio
The crucial first step is to find the common ratio (often denoted as 'r'). This is the constant value you multiply one term by to get the next term in the sequence. I determined the common ratio by:
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Dividing a term by its preceding term. If you have two consecutive terms, dividing the later term by the earlier term will yield the common ratio. For example, if the sequence contains the terms 4 and 8 consecutively, the common ratio is 8 / 4 = 2.
- Formula:
r = aₙ / aₙ₋₁(where aₙ is any term and aₙ₋₁ is the term before it)
- Formula:
Calculating Missing Terms
Once I established the common ratio, I used it to find the missing terms in one of two ways:
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Multiplying: If a missing term comes after a known term, I multiplied the known term by the common ratio to find the missing term.
- Example: Sequence: 2, 4, __ , 16. The common ratio is 2 (4/2 = 2). To find the missing term, I multiply 4 * 2 = 8. So, the missing term is 8.
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Dividing: If a missing term comes before a known term, I divided the known term by the common ratio to find the missing term.
- Example: Sequence: 2, __ , 8, 16. The common ratio is 2 (16/8 = 2). To find the missing term, I divide 8 / 2 = 4. So, the missing term is 4.
Summary Table
| Scenario | Operation | Example (r=3) |
|---|---|---|
| Missing Term After Known Term | Multiply | Sequence: 2, __ , 18. Missing Term = 2 3 = 6, then 6 3 = 18, check. |
| Missing Term Before Known Term | Divide | Sequence: 2, 6, . Common ratio calculated 6/2=3. Sequence: , 6. Missing Term = 6 / 3 = 2. |
By consistently applying these two operations (multiplication and division, based on the position of the missing term relative to the known terms) after correctly calculating the common ratio, I was able to determine the missing terms in any geometric sequence.
