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.