The image contains various problems related to geometric sequences. Let me extract and solve them one by one.

Problem 1: Determine if the sequence is geometric. If it is, find the common ratio.

• $11, 22, 44, 88, \ldots$

  • This sequence has a constant ratio between consecutive terms: $\frac{22}{11} = 2, \quad \frac{44}{22} = 2, \quad \frac{88}{44} = 2$ The sequence is geometric with a common ratio $r = 2$.

• $3, -6, 12, -24, \ldots$

  • Checking the ratio: $\frac{-6}{3} = -2, \quad \frac{12}{-6} = -2, \quad \frac{-24}{12} = -2$ The sequence is geometric with a common ratio $r = -2$.

Problem 2: Find the explicit formula for the geometric sequence.

• Given $a_1 = 4$ and $r = 3$, the general formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1} = 4 \cdot 3^{n-1}$

Problem 3: Find the first five terms and the sixth term.

• Using the formula $a_n = 4 \cdot 3^{n-1}$, let's compute: $a_1 = 4, \quad a_2 = 4 \cdot 3 = 12, \quad a_3 = 4 \cdot 9 = 36, \quad a_4 = 4 \cdot 27 = 108, \quad a_5 = 4 \cdot 81 = 324$ The first five terms are $4, 12, 36, 108, 324$.

Do you want me to solve the remaining problems? Here are some follow-up questions:

1. What is the common ratio for the sequence $5, 10, 20, 40, \ldots$?
2. How do you derive the explicit formula for any geometric sequence?
3. What is the significance of the first term in a geometric sequence?
4. Can a geometric sequence have a common ratio of 1? What does that imply?
5. How does a negative common ratio affect the terms of a geometric sequence?

Tip: Always verify the ratio between terms to confirm if a sequence is geometric.