what is the 5th term of 4,12,36,108 by using infinite geometric series

To find the 5th term of the sequence \( 4, 12, 36, 108, \dots \), we can identify this as a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, \( r \).

### Step 1: Identify the first term and common ratio
- **First term (\(a_1\))**: \( 4 \)
- **Common ratio (\(r\))**: To find \(r\), divide the second term by the first term:
  \[
  r = \frac{12}{4} = 3
  \]

### Step 2: Find the 5th term
The general formula for the \(n\)th term of a geometric sequence is:
\[
a_n = a_1 \times r^{n-1}
\]
For the 5th term (\(a_5\)):
\[
a_5 = 4 \times 3^{5-1} = 4 \times 3^4
\]
\[
a_5 = 4 \times 81 = 324
\]

### Answer:
The 5th term of the sequence is **324**.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How do you derive the formula for the \(n\)th term of a geometric sequence?
2. How does the sum of an infinite geometric series differ from finding a specific term?
3. What is the sum of the first 5 terms of this geometric series?
4. How would the sequence change if the common ratio was negative?
5. Can a geometric sequence have a common ratio of 1?

**Tip:** When working with geometric sequences, always double-check the common ratio by dividing consecutive terms to ensure consistency.

Discover how to find the 5th term of the geometric sequence 4, 12, 36, 108 using the formula for the nth term of a geometric sequence. This problem involves identifying the first term, common ratio, and applying the formula to calculate the 5th term. Suitable for high school students studying geometric sequences.

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