find the 25th term of the geometric sequence 1, 1/5, 1/25, 1/125, ....

To find the 25th term of a geometric sequence, we need to use the formula for the \(n\)-th term of a geometric sequence:

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

where:
- \(a_n\) is the \(n\)-th term.
- \(a_1\) is the first term of the sequence.
- \(r\) is the common ratio.
- \(n\) is the term number.

For the given sequence: \(1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \ldots\)

- The first term \(a_1 = 1\).
- The common ratio \(r\) can be found by dividing the second term by the first term: 

\[
r = \frac{\frac{1}{5}}{1} = \frac{1}{5}
\]

Now, to find the 25th term (\(a_{25}\)), we plug in the values into the formula:

\[
a_{25} = 1 \cdot \left(\frac{1}{5}\right)^{24}
\]

Let's compute this value.

\[
a_{25} = \left(\frac{1}{5}\right)^{24}
= \frac{1}{5^{24}}
\]

So, the 25th term of the geometric sequence is:

\[
a_{25} = \frac{1}{5^{24}}
\]

### Would you like more details, or do you have any questions?

### Related Questions:
1. What is the formula for the sum of the first \(n\) terms of a geometric sequence?
2. How do you find the common ratio of a geometric sequence if you're given two non-consecutive terms?
3. Can a geometric sequence have a common ratio greater than 1?
4. How does the behavior of a geometric sequence differ if the common ratio is negative?
5. How can geometric sequences be applied to real-world situations, such as calculating compound interest?

### Tip:
When dealing with very large exponents, it's useful to understand properties of powers and logarithms to simplify calculations and make sense of the magnitude of the numbers.

Find the 25th term of the geometric sequence 1, 1/5, 1/25, 1/125, ....

Discover how to find the 25th term of the geometric sequence starting with 1, 1/5, 1/25, 1/125, ... using the formula for the nth term of a geometric sequence. This problem covers key concepts in sequences and series, ideal for students in Grades 9-10.

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