The given series is: $4\sqrt{3}, \ \sqrt{3}, \ -2\sqrt{3}, \dots$

Let's determine the common ratio and check if this is a geometric sequence. In a geometric sequence, the ratio between consecutive terms is constant.

Step 1: Find the common ratio

To find the common ratio $r$, divide the second term by the first term:

$r = \frac{\sqrt{3}}{4\sqrt{3}} = \frac{1}{4}$

Now, verify the ratio by dividing the third term by the second term:

$r = \frac{-2\sqrt{3}}{\sqrt{3}} = -2$

As the ratios are not equal, this is not a geometric series.

Since the series is not geometric, let's check if it’s arithmetic by finding the common difference between terms.

Step 2: Check if it's an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the differences between the first two terms:

$d = \sqrt{3} - 4\sqrt{3} = -3\sqrt{3}$

Now, check the difference between the second and third terms:

$d = -2\sqrt{3} - \sqrt{3} = -3\sqrt{3}$

The common difference $d = -3\sqrt{3}$, so this is an arithmetic sequence with:

• First term $a = 4\sqrt{3}$
• Common difference $d = -3\sqrt{3}$

Step 3: Find the 12th term

The $n$-th term of an arithmetic sequence is given by the formula:

$a_n = a + (n-1) \cdot d$

Substitute the known values to find the 12th term $a_{12}$:

$a_{12} = 4\sqrt{3} + (12-1) \cdot (-3\sqrt{3})$

$a_{12} = 4\sqrt{3} + 11 \cdot (-3\sqrt{3})$

$a_{12} = 4\sqrt{3} - 33\sqrt{3}$

$a_{12} = -29\sqrt{3}$

Thus, the 12th term of the series is $-29\sqrt{3}$.

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Would you like further details on how to derive the formula, or have any other questions?

Here are 5 related questions for practice:

1. What is the 15th term of the same series?
2. How do you determine if a sequence is arithmetic or geometric?
3. Find the sum of the first 12 terms of this arithmetic sequence.
4. What is the common ratio in a geometric sequence, and how does it affect the sequence?
5. If the common difference were $-5\sqrt{3}$, what would the 12th term be?

Tip: Always verify the type of sequence (arithmetic or geometric) by checking either the difference or the ratio between terms.