Let's analyze both sequences in the image to determine if they are arithmetic or geometric.

Sequence a: $1, -2, 4, -8, \dots$

1. Check for Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.

   • $-2 - 1 = -3$
   • $4 - (-2) = 6$
   • $-8 - 4 = -12$

   Since the differences are not the same ($-3, 6, -12$), it is not an arithmetic sequence.

2. Check for Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.

   • $\frac{-2}{1} = -2$
   • $\frac{4}{-2} = -2$
   • $\frac{-8}{4} = -2$

   The ratio is consistently $-2$, so this is a geometric sequence with a common ratio of $-2$.

Sequence b: $1, -2, -5, -8, \dots$

1. Check for Arithmetic Sequence:

   • $-2 - 1 = -3$
   • $-5 - (-2) = -3$
   • $-8 - (-5) = -3$

   The difference between consecutive terms is consistently $-3$, so this is an arithmetic sequence with a common difference of $-3$.

2. Check for Geometric Sequence: We don't need to check for a geometric sequence since we've already confirmed it's arithmetic, but for completeness:

   • $\frac{-2}{1} = -2$
   • $\frac{-5}{-2} = 2.5$
   • $\frac{-8}{-5} = 1.6$

   The ratios are inconsistent, confirming it's not geometric.

Conclusion:

• Sequence (a) is geometric with a common ratio of $-2$.
• Sequence (b) is arithmetic with a common difference of $-3$.

Would you like further details or have any questions about these sequences?

Here are five related questions:

1. How do you identify the common ratio in a geometric sequence?
2. What is the general formula for the $n$-th term of an arithmetic sequence?
3. Can a sequence be both arithmetic and geometric? Why or why not?
4. How do you derive the sum of a finite geometric series?
5. What are some applications of arithmetic and geometric sequences in real life?

Tip: In a geometric sequence, negative common ratios result in alternating positive and negative terms!