Let's analyze each sequence to determine if it is geometric, and if so, find the common ratio.

(a) $16, 20, 24, 28, \dots$

To determine if the sequence is geometric, we check if there is a constant ratio between consecutive terms. We can do this by dividing each term by the previous term:

$\frac{20}{16} = 1.25, \quad \frac{24}{20} = 1.2, \quad \frac{28}{24} \approx 1.167$

Since the ratios are not constant, the sequence is not geometric.

(b) $4, -12, 36, -108, \dots$

Check the ratio between consecutive terms:

$\frac{-12}{4} = -3, \quad \frac{36}{-12} = -3, \quad \frac{-108}{36} = -3$

Since the ratio is constant, the sequence is geometric with a common ratio $r = -3$.

(c) $25, 5, 1, \frac{1}{5}, \dots$

Check the ratio between consecutive terms:

$\frac{5}{25} = \frac{1}{5}, \quad \frac{1}{5} = \frac{1}{5}, \quad \frac{\frac{1}{5}}{1} = \frac{1}{5}$

Since the ratio is constant, the sequence is geometric with a common ratio $r = \frac{1}{5}$.

Here is the summary:

• (a) Not geometric.
• (b) Geometric with $r = -3$.
• (c) Geometric with $r = \frac{1}{5}$.

Would you like more details on any of the steps?

Follow-up questions:

1. How can we determine if a sequence is arithmetic instead of geometric?
2. Can a sequence be both arithmetic and geometric?
3. What is the difference between common difference and common ratio in sequences?
4. How do geometric series relate to geometric sequences?
5. What is the formula for the sum of a geometric series?

Tip: In a geometric sequence, the ratio between consecutive terms must remain constant throughout.