How many terms are less than 10000
in the geometric sequence 6,24,96?

The geometric sequence given is \(6, 24, 96, \ldots\).

First, let's identify the first term \(a\) and the common ratio \(r\):

- \(a = 6\)
- \(r = \frac{24}{6} = 4\)

The general term of a geometric sequence can be written as:
\[
a_n = a \cdot r^{n-1}
\]
Substituting the values for \(a\) and \(r\):
\[
a_n = 6 \cdot 4^{n-1}
\]
We need to find the number of terms for which \(a_n < 10000\). So, we set up the inequality:
\[
6 \cdot 4^{n-1} < 10000
\]
Dividing both sides by 6:
\[
4^{n-1} < \frac{10000}{6} \approx 1666.67
\]
Now, take the logarithm (base 10 or base 4) of both sides:
\[
\log_{10}(4^{n-1}) < \log_{10}(1666.67)
\]
Using the logarithm power rule \( \log_b(x^y) = y \log_b(x) \), we get:
\[
(n-1) \log_{10}(4) < \log_{10}(1666.67)
\]
\[
n-1 < \frac{\log_{10}(1666.67)}{\log_{10}(4)}
\]
\[
n-1 < \frac{\log_{10}(1666.67)}{0.60206} \approx \frac{3.2221}{0.60206} \approx 5.35
\]
Since \(n-1\) must be an integer, \(n-1 \leq 5\). Therefore, \(n \leq 6\).

So, the number of terms \(n\) is 6.

### Final Answer:
There are **6** terms in the sequence that are less than 10000.

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Would you like more details or have any questions? Here are some related questions:

1. How do you calculate the sum of the first 6 terms in this sequence?
2. What if the common ratio was different? How would it affect the number of terms?
3. How would the approach change if you had to find terms less than a different number, say 50000?
4. Can you explain how logarithms help solve exponential inequalities?
5. What is the relationship between geometric sequences and exponential functions?

**Tip:** When working with geometric sequences, identifying the common ratio is key to solving most related problems.

Learn how to calculate the number of terms less than 10000 in the geometric sequence 6, 24, 96 using the general term formula and logarithms. Explore the application of exponential inequalities and logarithm rules in solving this problem. Suitable for high school students studying geometric sequences.

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