## Introduction

Welcome to our comprehensive answer key for Secondary Math 1 Module 1 Sequences 1.3. In this module, we will explore various types of sequences and their properties. By understanding the answer key, you can deepen your knowledge and improve your problem-solving skills in the subject of mathematics. So, let's dive right in!

## What is Secondary Math 1 Module 1 Sequences 1.3?

Secondary Math 1 Module 1 Sequences 1.3 is a section of the curriculum that focuses on sequences in mathematics. In this particular module, we will be looking at various aspects of sequences, including the types of sequences, finding terms in a sequence, and determining the nth term of a sequence. This module is an essential building block for future mathematical concepts and is crucial for a solid understanding of the subject.

### Types of Sequences

1. Arithmetic Sequences

2. Geometric Sequences

3. Recursive Sequences

### Finding Terms in a Sequence

1. The Explicit Formula

2. The Recursive Formula

3. Using Given Information

### Determining the nth Term of a Sequence

1. Using the Explicit Formula

2. Using the Recursive Formula

3. Analyzing the Pattern

## Answer Key for Secondary Math 1 Module 1 Sequences 1.3

### Arithmetic Sequences

Arithmetic sequences are sequences in which the difference between consecutive terms is constant. To find the nth term of an arithmetic sequence, we can use the formula:

a_n = a_1 + (n-1)d

Where:

a_n is the nth term

a_1 is the first term

n is the position of the term

d is the common difference

Example:

Find the 10th term of the arithmetic sequence: 2, 5, 8, 11, ...

Using the formula, we have:

a_10 = 2 + (10-1)3

a_10 = 2 + 27

a_10 = 29

Therefore, the 10th term of the sequence is 29.

### Geometric Sequences

Geometric sequences are sequences in which the ratio between consecutive terms is constant. To find the nth term of a geometric sequence, we can use the formula:

a_n = a_1 * r^(n-1)

Where:

a_n is the nth term

a_1 is the first term

r is the common ratio

Example:

Find the 7th term of the geometric sequence: 2, 6, 18, 54, ...

Using the formula, we have:

a_7 = 2 * (3)^(7-1)

a_7 = 2 * 3^6

a_7 = 2 * 729

a_7 = 1458

Therefore, the 7th term of the sequence is 1458.

### Recursive Sequences

Recursive sequences are sequences in which each term is defined in terms of the previous terms. To find the nth term of a recursive sequence, we need to use the recursive formula.

Example:

Consider the following recursive sequence:

a_1 = 2

a_n = a_(n-1) + 3

To find the 5th term of this sequence, we can use the recursive formula:

a_5 = a_4 + 3

a_5 = (a_3 + 3) + 3

a_5 = ((a_2 + 3) + 3) + 3

a_5 = (((a_1 + 3) + 3) + 3) + 3

a_5 = (((2 + 3) + 3) + 3) + 3

a_5 = 14

Therefore, the 5th term of the sequence is 14.

### Finding Terms in a Sequence

There are several methods to find terms in a sequence, depending on the given information and the type of sequence.

1. The Explicit Formula:

If the sequence is arithmetic or geometric, we can use the explicit formula to find any term in the sequence. The explicit formula provides a direct way to calculate the nth term without needing to find all the previous terms.

2. The Recursive Formula:

If the sequence is recursive, we need to use the recursive formula to find any term in the sequence. The recursive formula defines each term in terms of the previous terms, so we need to find all the previous terms before finding the desired term.

3. Using Given Information:

Sometimes, the problem may provide specific information about the sequence, such as the first term, the last term, or the sum of the terms. In such cases, we can use this given information to find the desired term.

### Determining the nth Term of a Sequence

To determine the nth term of a sequence, we need to analyze the pattern and use the appropriate formula.

1. Using the Explicit Formula:

If the sequence is arithmetic or geometric, we can use the explicit formula to determine the nth term. By plugging in the values of the first term, the common difference or ratio, and the position of the term, we can find the desired term.

2. Using the Recursive Formula:

If the sequence is recursive, we need to use the recursive formula to determine the nth term. By recursively applying the formula until we reach the desired term, we can find the nth term of the sequence.

3. Analyzing the Pattern:

Sometimes, the pattern in the sequence may not follow a specific formula. In such cases, we need to carefully analyze the pattern and find a rule or formula that describes the relationship between the terms. This may require some creativity and logical thinking.

## Conclusion

By understanding the answer key for Secondary Math 1 Module 1 Sequences 1.3, you can enhance your skills in working with sequences. Whether it's arithmetic, geometric, or recursive sequences, the key is to practice and familiarize yourself with the different formulas and methods. So, keep exploring and mastering the world of sequences in mathematics!