(a) Sketch x[n] = a
n
for a typical a in the range -1 < a < 0.

Here is the sketch of the discrete-time sequence \( x[n] = a^n \) for a typical value of \( a \) in the range \( -1 < a < 0 \). The graph shows the expected alternating sign and exponential decay pattern.

Would you like further details or have any questions?

Here are five related questions to explore:
1. How does the behavior of \( x[n] = a^n \) change if \( a \) is positive and less than 1?
2. What happens to the sequence if \( a \) is greater than 1?
3. How does the oscillation pattern change as \( a \) approaches -1?
4. Can you explain the role of \( a \) in controlling the decay rate of the sequence?
5. How would the graph look if \( a \) were exactly -1?

**Tip**: For sequences with negative bases, always consider the even and odd powers to understand sign changes.

Sketch x[n] = a^n for a typical a in the range -1 < a < 0.

Explore how to sketch the discrete-time sequence x[n] = a^n for values of a between -1 and 0. Understand the alternating sign and exponential decay pattern, and discover how the value of a affects the sequence's behavior. Suitable for high school students studying discrete-time signals and sequences.

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