Find the 12th term of the series 4√3, √3, -2√3

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 \( -29\sqrt{3} \).

---

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.

Learn how to find the 12th term of an arithmetic sequence with the terms 4√3, √3, -2√3. This detailed solution walks through identifying the common difference and applying the n-th term formula. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation