Given the sequence: 
\[ 4\sqrt{3}, 9\sqrt{3}, 14\sqrt{3}, 19\sqrt{3} \]

we need to find the general rule or nth term of this sequence.

### Analyzing the Sequence

Let's extract the coefficients of \(\sqrt{3}\):
\[ 4, 9, 14, 19 \]

This sequence is:
\[ 4, 9, 14, 19 \]

This sequence is an arithmetic sequence with the first term \(a = 4\) and the common difference \(d = 5\).

### General Term of the Arithmetic Sequence

The nth term of an arithmetic sequence is given by:
\[ a_n = a + (n-1)d \]

Substituting the values for \(a\) and \(d\):
\[ a_n = 4 + (n-1) \cdot 5 \]
\[ a_n = 4 + 5n - 5 \]
\[ a_n = 5n - 1 \]

### Incorporating \(\sqrt{3}\)

Since each term of the sequence is multiplied by \(\sqrt{3}\), we multiply the general term of the arithmetic sequence by \(\sqrt{3}\):
\[ T_n = (5n - 1)\sqrt{3} \]

### General Rule

Therefore, the general rule for the nth term of the sequence is:
\[ T_n = (5n - 1)\sqrt{3} \]

Would you like more details or have any questions?

Here are 8 related questions:
1. How do you determine the common difference in an arithmetic sequence?
2. What is the general formula for the nth term of an arithmetic sequence?
3. Can you explain why multiplying the sequence by \(\sqrt{3}\) affects the general term?
4. How would the general term change if the common difference were different?
5. Can you give an example of a geometric sequence?
6. What are some methods to identify the type of sequence given a list of numbers?
7. How can you verify if your general rule for the nth term is correct?
8. What are some real-life applications of arithmetic sequences?

**Tip:** When dealing with sequences, always look for patterns in differences or ratios to identify the type of sequence (arithmetic, geometric, etc.). This will help in deriving the correct formula for the nth term.

Discover how to find the general term of the sequence 4√3, 9√3, 14√3, 19√3 with a step-by-step solution. This problem involves arithmetic sequences and multiplication by √3. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation