To solve this problem, we need to identify the 10th term of the given geometric sequence.

Step 1: Identify the first term and common ratio

The given sequence is:

$10, 2, \frac{2}{5}, \frac{2}{25}, \frac{2}{125}, \ldots$

• The first term ($a$) is $10$.
• To find the common ratio ($r$), we divide the second term by the first term:

$r = \frac{2}{10} = \frac{1}{5}.$

Step 2: Use the formula for the nth term of a geometric sequence

The formula for the $n$th term of a geometric sequence is:

$a_n = a \cdot r^{n-1},$

where:

• $a$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

Step 3: Find the 10th term

Substitute $a = 10$, $r = \frac{1}{5}$, and $n = 10$:

$a_{10} = 10 \cdot \left(\frac{1}{5}\right)^{10-1} = 10 \cdot \left(\frac{1}{5}\right)^9.$

Now, calculate $\left(\frac{1}{5}\right)^9$:

$\left(\frac{1}{5}\right)^9 = \frac{1}{5^9} = \frac{1}{1953125}.$

So, the 10th term is:

$a_{10} = 10 \cdot \frac{1}{1953125} = \frac{10}{1953125}.$

Simplify this fraction:

$\frac{10}{1953125} = \frac{2}{390625}.$

Answer:

The 10th term is $\frac{2}{390625}$, which corresponds to option A.

Do you have any questions or would you like further details?

Here are some related questions for deeper understanding:

1. How do you determine if a sequence is geometric?
2. How do you find the sum of a finite geometric sequence?
3. What is the difference between arithmetic and geometric sequences?
4. How do you solve for the common ratio if it is unknown?
5. Can you find the 15th term of this sequence?

Tip: Always verify the common ratio by dividing consecutive terms to ensure the sequence is truly geometric.