### Understanding Sequences in Mathematics#### Problem Statement:"Describe what happens to the sequence after the 12th term."In this exercise, you are required to analyze a given sequence in mathematics and describe its behavior after the 12th term. A clear understanding of sequences and their properties is essential to solve this type of problem accurately.**Steps to approach this problem:**1. **Identify the sequence pattern:** - Study the first few terms of the sequence to understand the pattern or rule governing it. - Determine if the sequence is arithmetic, geometric, or follows a different recognizable pattern.2. **Establish the general term:** - Write out a formula or general term (nth term) that represents the sequence. - This formula will help in predicting any term in the sequence, including those beyond the 12th term.3. **Analyze the 12th term:** - Calculate or identify the 12th term using the general term or pattern you have established. - Ensure that all preceding terms are consistent with the identified rule.4. **Predict subsequent behavior:** - Use the general term to describe the behavior of the sequence beyond the 12th term. - Consider conditions like convergence or divergence if the sequence is infinite, or identify the terminus if it is finite.**Example**:If the sequence is given by an arithmetic formula $ a_n = 3n + 1 $:- The 12th term is $ a_{12} = 3(12) + 1 = 37 $.- The behavior after the 12th term continues to increase by 3 units per subsequent term.By thoroughly analyzing the sequence and implementing the aforementioned steps, you will be able to describe with clarity what happens to the sequence after the 12th term. ### Sequence Analysis Problem**Given the following sequence:**\[ 33, 30, 27, 24, \ldots \]**[2] Write a formula to generate the $ n^{\text{th}} $ term and be sure to include the necessary parameters.**---In this problem, you are provided with a mathematical sequence and asked to derive a formula to describe the $ n^{\text{th}} $ term in that sequence. The sequence provided is:\[ 33, 30, 27, 24, \ldots \]Observe that each term in the sequence decreases by 3. This suggests a linear pattern where a constant difference (also called the common difference) is subtracted from each term to get the next term.To generate the $ n^{\text{th}} $ term of this arithmetic sequence, we can use the standard formula for the $ n^{\text{th}} $ term of an arithmetic sequence, which is given by:\[ a_n = a_1 + (n-1)d \]where:- $ a_n $ is the $ n^{\text{th}} $ term of the sequence,- $ a_1 $ is the first term of the sequence,- $ d $ is the common difference,- $ n $ is the term number.For the given sequence:- The first term $ a_1 $ is 33,- The common difference $ d $ is -3 (since the sequence decreases by 3 each time).Therefore, the formula to generate the $ n^{\text{th}} $ term of the sequence is:\[ a_n = 33 + (n-1)(-3) \]Simplifying the formula:\[ a_n = 33 - 3(n-1) \]\[ a_n = 33 - 3n + 3 \]\[ a_n = 36 - 3n \]So, the $ n^{\text{th}} $ term is given by:\[ a_n = 36 - 3n \]Using this formula, you can calculate the value of any term in the sequence$ a_n $ by substituting the value of $ n $.

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Description: ### Understanding Sequences in Mathematics#### Problem Statement:"Describe what happens to the sequence after the 12th term."In this exercise, you are required to analyze a given sequence in mathematics and describe its behavior after the 12th term.

Answered: Given the following sequence: 33, 30,…

Math

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Given the following sequence: 33, 30, 27, 24, ... [2] Write a formula to generate the nh term and be sure to include the necessary parameters.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

### Understanding Sequences in Mathematics

#### Problem Statement:
"Describe what happens to the sequence after the 12th term."

In this exercise, you are required to analyze a given sequence in mathematics and describe its behavior after the 12th term. A clear understanding of sequences and their properties is essential to solve this type of problem accurately.

**Steps to approach this problem:**

1. **Identify the sequence pattern:**
- Study the first few terms of the sequence to understand the pattern or rule governing it.
- Determine if the sequence is arithmetic, geometric, or follows a different recognizable pattern.

2. **Establish the general term:**
- Write out a formula or general term (nth term) that represents the sequence.
- This formula will help in predicting any term in the sequence, including those beyond the 12th term.

3. **Analyze the 12th term:**
- Calculate or identify the 12th term using the general term or pattern you have established.
- Ensure that all preceding terms are consistent with the identified rule.

4. **Predict subsequent behavior:**
- Use the general term to describe the behavior of the sequence beyond the 12th term.
- Consider conditions like convergence or divergence if the sequence is infinite, or identify the terminus if it is finite.

**Example**:
If the sequence is given by an arithmetic formula $ a_n = 3n + 1 $:
- The 12th term is $ a_{12} = 3(12) + 1 = 37 $.
- The behavior after the 12th term continues to increase by 3 units per subsequent term.

By thoroughly analyzing the sequence and implementing the aforementioned steps, you will be able to describe with clarity what happens to the sequence after the 12th term.

### Sequence Analysis Problem

**Given the following sequence:**

\[ 33, 30, 27, 24, \ldots \]

**[2] Write a formula to generate the $ n^{\text{th}} $ term and be sure to include the necessary parameters.**

---

In this problem, you are provided with a mathematical sequence and asked to derive a formula to describe the $ n^{\text{th}} $ term in that sequence. The sequence provided is:

\[ 33, 30, 27, 24, \ldots \]

Observe that each term in the sequence decreases by 3. This suggests a linear pattern where a constant difference (also called the common difference) is subtracted from each term to get the next term.

To generate the $ n^{\text{th}} $ term of this arithmetic sequence, we can use the standard formula for the $ n^{\text{th}} $ term of an arithmetic sequence, which is given by:

\[ a_n = a_1 + (n-1)d \]

where:
- $ a_n $ is the $ n^{\text{th}} $ term of the sequence,
- $ a_1 $ is the first term of the sequence,
- $ d $ is the common difference,
- $ n $ is the term number.

For the given sequence:
- The first term $ a_1 $ is 33,
- The common difference $ d $ is -3 (since the sequence decreases by 3 each time).

Therefore, the formula to generate the $ n^{\text{th}} $ term of the sequence is:

\[ a_n = 33 + (n-1)(-3) \]

Simplifying the formula:

\[ a_n = 33 - 3(n-1) \]
\[ a_n = 33 - 3n + 3 \]
\[ a_n = 36 - 3n \]

So, the $ n^{\text{th}} $ term is given by:

\[ a_n = 36 - 3n \]

Using this formula, you can calculate the value of any term in the sequence$ a_n $ by substituting the value of $ n $.

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