Introduction
Welcome to our comprehensive answer key for Secondary Math 1 Module 1 Sequences 1.5. In this article, we will provide you with detailed solutions and explanations for all the questions in this module. By using this answer key, you can enhance your understanding of sequences and improve your problem-solving skills. Let's dive in!
Understanding Sequences
Definition of Sequences
Before we begin, let's define what a sequence is in the context of mathematics. A sequence is an ordered list of numbers in which each term is determined by a specific rule or pattern. These patterns can be arithmetic, geometric, or even more complex. By studying sequences, we can uncover fascinating relationships between numbers and explore the underlying patterns.
Types of Sequences
Sequences can be classified into different types based on their patterns. The main types of sequences we encounter are arithmetic sequences and geometric sequences.
Arithmetic Sequences
In an arithmetic sequence, each term is obtained by adding a fixed number, called the common difference, to the previous term. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3. To find the nth term of an arithmetic sequence, we can use the formula: a_n = a_1 + (n-1)d, where a_n represents the nth term, a_1 is the first term, and d is the common difference.
Geometric Sequences
Geometric sequences, on the other hand, have a common ratio between consecutive terms. Each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. For example, the sequence 3, 6, 12, 24, 48 is a geometric sequence with a common ratio of 2. The formula to find the nth term of a geometric sequence is: a_n = a_1 * r^(n-1), where a_n represents the nth term, a_1 is the first term, and r is the common ratio.
The Answer Key
Question 1
Question: Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...
Solution: To find the 10th term of the arithmetic sequence, we can use the formula a_n = a_1 + (n-1)d. In this case, the first term (a_1) is 3 and the common difference (d) is 4. Plugging these values into the formula, we get: a_10 = 3 + (10-1)4 = 3 + 9*4 = 3 + 36 = 39. Therefore, the 10th term of the sequence is 39.
Question 2
Question: Find the sum of the first 15 terms of the geometric sequence 2, 6, 18, 54, ...
Solution: To find the sum of the first 15 terms of a geometric sequence, we can use the formula S_n = a_1 * (1 - r^n) / (1 - r), where S_n represents the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms. In this case, the first term (a_1) is 2, the common ratio (r) is 3, and the number of terms (n) is 15. Plugging these values into the formula, we get: S_15 = 2 * (1 - 3^15) / (1 - 3) = 2 * (-14348906) / (-2) = 14348906. Therefore, the sum of the first 15 terms of the sequence is 14348906.
Question 3
Question: Find the missing term in the arithmetic sequence 2, 5, __, 11, 14, 17.
Solution: To find the missing term in an arithmetic sequence, we need to identify the common difference and find the term that satisfies the pattern. In this case, the common difference is 3. To find the missing term, we can look at the pattern of the sequence. The missing term should be 8. Therefore, the sequence is 2, 5, 8, 11, 14, 17.
Conclusion
By using this answer key for Secondary Math 1 Module 1 Sequences 1.5, you can reinforce your understanding of sequences and improve your problem-solving abilities. Remember to study the patterns and formulas for arithmetic and geometric sequences, as they will come in handy when solving similar problems. Practice makes perfect, so keep challenging yourself with more sequences and enjoy the beauty of mathematics!