Let's break down the equation you provided:

$a_n = 2a_{n-1} + a_{n-2} - 2a_{n-3}$

This is a recursive relation, which defines each term $a_n$ in terms of the previous three terms in the sequence: $a_{n-1}, a_{n-2},$ and $a_{n-3}$.

You also provided initial conditions:

• $a_0 = 3$
• $a_1 = 6$
• $a_2 = 0$

To find further terms in the sequence (e.g., $a_3, a_4$, etc.), we can apply the recursive relation step by step.

Let's find $a_3$:

Using the equation $a_n = 2a_{n-1} + a_{n-2} - 2a_{n-3}$ for $n = 3$:

$a_3 = 2a_2 + a_1 - 2a_0$

Substituting the known values $a_2 = 0$, $a_1 = 6$, and $a_0 = 3$:

$a_3 = 2(0) + 6 - 2(3)$

$a_3 = 6 - 6 = 0$

So, $a_3 = 0$.

Let's find $a_4$:

Now, for $n = 4$:

$a_4 = 2a_3 + a_2 - 2a_1$

Substitute the known values $a_3 = 0$, $a_2 = 0$, and $a_1 = 6$:

$a_4 = 2(0) + 0 - 2(6)$

$a_4 = 0 - 12 = -12$

So, $a_4 = -12$.

Let's find $a_5$:

Now, for $n = 5$:

$a_5 = 2a_4 + a_3 - 2a_2$

Substitute the known values $a_4 = -12$, $a_3 = 0$, and $a_2 = 0$:

$a_5 = 2(-12) + 0 - 2(0)$

$a_5 = -24 + 0 = -24$

So, $a_5 = -24$.

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Would you like to see more terms of the sequence or a deeper explanation?

Related questions:

1. How do we determine the general form of a recursive sequence?
2. Can we find an explicit formula for this recursive sequence?
3. How does changing initial conditions affect the sequence?
4. How are recursive sequences used in real-world problems?
5. What happens if we add more terms to the recursive equation?

Tip:

When working with recursive relations, having clear initial conditions is essential to generating the correct sequence. Always verify each step carefully to avoid errors when calculating subsequent terms!