Which term is the first to have a value less than 20 in the recursive sequence 256, 192, 144, 108

The given sequence is \( 256, 192, 144, 108, \dots \).

This appears to be a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio. Let's verify this and calculate the ratio:

\[
\frac{192}{256} = \frac{3}{4}, \quad \frac{144}{192} = \frac{3}{4}, \quad \frac{108}{144} = \frac{3}{4}
\]

Thus, the common ratio is \( \frac{3}{4} \).

The general formula for the \(n\)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

Where:
- \(a_1 = 256\) (the first term),
- \(r = \frac{3}{4}\) (the common ratio),
- \(n\) is the term number.

We are asked to find the first term that is less than 20, so we want to solve for \(n\) when \(a_n < 20\):

\[
256 \cdot \left(\frac{3}{4}\right)^{n-1} < 20
\]

Now, let's solve for \(n\).

\[
\left(\frac{3}{4}\right)^{n-1} < \frac{20}{256} = \frac{5}{64}
\]

Taking the natural logarithm on both sides:

\[
(n-1) \ln\left(\frac{3}{4}\right) < \ln\left(\frac{5}{64}\right)
\]

\[
(n-1) < \frac{\ln\left(\frac{5}{64}\right)}{\ln\left(\frac{3}{4}\right)}
\]

Let me calculate that.The solution gives \(n \approx 9.86\), which means the 10th term will be the first to have a value less than 20.

Would you like to see the exact value of the 10th term or further details about the steps?

Here are 5 related questions:
1. How do you find the sum of the first \(n\) terms of a geometric sequence?
2. What is the behavior of a geometric sequence as \(n\) approaches infinity when \( |r| < 1 \)?
3. How would the result change if the ratio were larger than 1?
4. Can we solve a similar problem with an arithmetic sequence?
5. How do logarithms help in solving exponential inequalities?

**Tip:** When the common ratio is between 0 and 1, the terms in a geometric sequence decrease exponentially.

Which term is the first to have a value less than 20 in the recursive sequence 256, 192, 144, 108?

Discover how to find the first term in the geometric sequence 256, 192, 144, 108 that is less than 20. Learn the key concepts of geometric sequences, exponential decay, and how to solve exponential inequalities using logarithms. This problem is ideal for high school students in Grades 9-12.

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