How to find missing terms in a geometric sequence?
Understand the Problem
The question is asking how to determine the terms that are absent in a geometric sequence. A geometric sequence has a constant ratio between consecutive terms, so we will need to use that ratio to find the missing terms.
Answer
$ a_n = a \cdot r^{(n-1)} $
Answer for screen readers
The answer will vary based on the specific terms given in the geometric sequence and the missing values. Once the first term and common ratio are identified and used in the formula, each missing term can be solved accordingly.
Steps to Solve
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Identify the first term and the common ratio
In a geometric sequence, the first term is usually denoted as $a$. The common ratio $r$ can be found by dividing any term by its preceding term:
$$ r = \frac{a_{2}}{a_{1}} $$
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Write the formula for the nth term
The formula for the nth term of a geometric sequence is given by:
$$ a_n = a \cdot r^{(n-1)} $$
Here, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.
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Locate the missing term positions
Identify which terms of the sequence are missing and their respective positions, usually denoted as $n$ in the formula.
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Use the formula to calculate missing terms
Substitute the first term $a$, the common ratio $r$, and the positions of the missing terms into the formula to compute their values:
$$ a_m = a \cdot r^{(m-1)} $$
for each missing position $m$.
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Verify the calculations
Check if the calculated values maintain the constant ratio with the surrounding terms in the sequence to confirm they're correct.
The answer will vary based on the specific terms given in the geometric sequence and the missing values. Once the first term and common ratio are identified and used in the formula, each missing term can be solved accordingly.
More Information
Determining missing terms in a geometric sequence helps reinforce understanding of sequences and series in mathematics. Geometric sequences are widely seen in exponential growth scenarios, like population growth or finance.
Tips
- Forgetting to maintain the common ratio when calculating consecutive terms.
- Misidentifying the first term $a$ or the common ratio $r$.
- Confusing the formula for an arithmetic sequence with a geometric sequence.
