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.

= \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.