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To find the 9th term of the given geometric sequence, let's analyze the sequence first:

Given sequence: \( 4, 20, 100, 500, \ldots \)

### Step 1: Identify the first term (\(a\)) and the common ratio (\(r\)).

- The first term \(a = 4\).
- The common ratio \(r\) can be found by dividing the second term by the first term:
  \[
  r = \frac{20}{4} = 5
  \]

### Step 2: Use the formula for the \(n\)th 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:
- \(a_n\) is the \(n\)th term,
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

### Step 3: Calculate the 9th term.

Substituting the values into the formula:
\[
a_9 = 4 \cdot 5^{(9-1)} = 4 \cdot 5^8
\]
Calculate \(5^8\):
\[
5^8 = 390625
\]
Now, calculate \(a_9\):
\[
a_9 = 4 \cdot 390625 = 1562500
\]

### Final Answer:
The 9th term of the geometric sequence is **1,562,500**.

Would you like more details or have any questions?

### Related Questions:
1. How would you find the sum of the first 9 terms of this geometric sequence?
2. What is the formula to find the sum of an infinite geometric series?
3. If the common ratio was negative, how would the sequence behave?
4. How can geometric sequences be used in real-life applications?
5. What is the 10th term of this sequence?

### Tip:
Remember that the common ratio in a geometric sequence is the factor that you multiply by to go from one term to the next.

Find the 9th term of the following geometric sequence: 4, 20, 100, 500, ...

Learn how to find the 9th term of a geometric sequence starting with 4, 20, 100, and 500. This step-by-step solution uses key algebraic concepts and the formula for the nth term of a geometric sequence. Ideal for students in Grades 9-12.

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