To find the common ratio of a geometric sequence with missing terms, you'll need to focus on the terms that are present and use them to deduce the ratio. This is how you'll calculate the common ratio and then use that information to find any missing terms.
Steps to Find the Common Ratio and Missing Terms
Here's a breakdown of the process, incorporating the provided reference information:
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Find the common ratio between known terms:
- If you have two consecutive terms, divide any term by the term that immediately precedes it. This quotient is your common ratio (r). Step 1: Find the common ratio of each pair of consecutive terms in the sequence by dividing each term by the term that came before it.
- If you have non-consecutive terms, calculate the difference in term positions. For example, if you know the 2nd and 5th term, the difference is 3. Use the nth root function to find the common ratio.
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Apply the common ratio to find missing terms:
- Once you've established the common ratio, multiply the term before the first missing term by the common ratio. Step 2: Multiply the common ratio with the number prior to the first missing number in the sequence. This will give you the next term.
- Repeat this process for any subsequent missing terms. Step 3: Repeat Step 2 for any other missing numbers.
Examples
Let's illustrate this with a few examples:
Example 1: Consecutive Known Terms
Sequence: 2, 6, 18, ___, 162
- Find the Common Ratio: 6/2 = 3 (or 18/6 = 3)
- Find the missing term: 18 * 3 = 54
- Full sequence: 2, 6, 18, 54, 162
Example 2: Non-Consecutive Known Terms
Sequence: 3, , , 24, ___, 192
- Identify known terms: The first term (a1) is 3 and the fourth term (a4) is 24.
- Calculate the difference in positions: 4 - 1 = 3
- Find the common ratio: To find the common ratio between the 1st and 4th term use:
- a4 = a1 * r^(4-1)
- 24 = 3 * r^3
- 8 = r^3
- r = 2
- Find the missing terms:
- 3 * 2 = 6
- 6 * 2 = 12
- 24 * 2 = 48
- Full Sequence: 3, 6, 12, 24, 48, 192
Summary
The key is to first establish the common ratio using the available terms and then apply it repeatedly to calculate any missing values.
