To find the first five terms of the sequence where:

$a_1 = -1$ $a_n = (-4)^{n-1} a_{n-1} - 3, \text{ for } n \geq 2,$

we calculate each term step by step:

Step 1: Calculate $a_2$

$a_2 = (-4)^{2-1} a_1 - 3$ $a_2 = (-4)^1 \cdot (-1) - 3$ $a_2 = 4 - 3 = 1$

Step 2: Calculate $a_3$

$a_3 = (-4)^{3-1} a_2 - 3$ $a_3 = (-4)^2 \cdot 1 - 3$ $a_3 = 16 - 3 = 13$

Step 3: Calculate $a_4$

$a_4 = (-4)^{4-1} a_3 - 3$ $a_4 = (-4)^3 \cdot 13 - 3$ $a_4 = -64 \cdot 13 - 3 = -832 - 3 = -835$

Step 4: Calculate $a_5$

$a_5 = (-4)^{5-1} a_4 - 3$ $a_5 = (-4)^4 \cdot (-835) - 3$ $a_5 = 256 \cdot (-835) - 3 = -213760 - 3 = -213763$

First Five Terms

The first five terms of the sequence are: $a_1 = -1, \; a_2 = 1, \; a_3 = 13, \; a_4 = -835, \; a_5 = -213763$

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Do you want a detailed explanation of how each term was calculated or have any further questions?

Five Related Questions:

1. What happens to the sequence for larger values of $n$?
2. How does the term $(-4)^{n-1}$ affect the growth of the sequence?
3. Can we derive a closed-form expression for this sequence?
4. What is the nature of the sequence (divergent, bounded, etc.)?
5. How does the sequence change if $a_1$ is a different value?

Tip:

When working with recursive sequences, always compute a few terms explicitly to understand the behavior before trying to generalize.