unit 10 sequences and series homework 1 answers

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02-02-2025

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Unit 10: Sequences and Series - Homework 1 Answers: A Comprehensive Guide

This guide provides detailed answers and explanations for the problems typically found in a Unit 10 homework assignment covering sequences and series. Since I don't have access to your specific homework sheet, I'll cover common problem types encountered in this unit. Remember to always refer to your textbook and class notes for specific definitions and formulas relevant to your course.

Understanding Sequences and Series:

Before diving into the problems, let's establish a foundational understanding. A sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence. We'll focus on two main types: arithmetic and geometric sequences and series.

1. Arithmetic Sequences and Series:

• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant (called the common difference, d). The nth term is given by: a_n = a₁ + (n-1)d, where a₁ is the first term.

• Arithmetic Series: The sum of an arithmetic sequence. The sum of the first n terms (S_n) is given by: S_n = n/2 [2a₁ + (n-1)d] or S_n = n/2(a₁ + a_n).

Example Problem (Arithmetic):

Find the 10th term and the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11...

Solution:

• Common difference (d): 5 - 2 = 3
• First term (a₁): 2
• 10th term (a₁₀): a₁₀ = 2 + (10-1)3 = 29
• Sum of the first 10 terms (S₁₀): S₁₀ = 10/2 (2(2) + (10-1)3) = 155 or S₁₀ = 10/2 (2 + 29) = 155

2. Geometric Sequences and Series:

• Geometric Sequence: A sequence where the ratio between consecutive terms is constant (called the common ratio, r). The nth term is given by: a_n = a₁ * r^(n-1)

• Geometric Series: The sum of a geometric sequence. The sum of the first n terms (S_n) is given by: S_n = a₁(1 - rⁿ) / (1 - r), where r ≠ 1. The sum of an infinite geometric series (|r| < 1) is given by: S_∞ = a₁ / (1 - r)

Example Problem (Geometric):

Find the 7th term and the sum of the first 7 terms of the geometric sequence 3, 6, 12, 24...

Solution:

• Common ratio (r): 6 / 3 = 2
• First term (a₁): 3
• 7th term (a₇): a₇ = 3 * 2^(7-1) = 192
• Sum of the first 7 terms (S₇): S₇ = 3(1 - 2⁷) / (1 - 2) = 381

3. Other Common Problem Types:

• Finding the nth term given certain terms: Use the formulas for arithmetic and geometric sequences to solve for the unknown variables (a₁, d, or r).
• Determining if a sequence is arithmetic or geometric: Calculate the differences or ratios between consecutive terms.
• Finding the sum of a finite or infinite series: Apply the appropriate formulas based on whether the series is arithmetic or geometric.
• Series involving sigma notation: Understand how to expand and evaluate summations using the formulas for arithmetic and geometric series.

Tips for Success:

• Clearly identify the type of sequence: Is it arithmetic, geometric, or neither?
• Write out the formulas: Having the correct formulas readily available will save time and reduce errors.
• Show your work: This helps you catch mistakes and allows for easier understanding of your solutions.
• Check your answers: Verify your results by plugging values back into the formulas or using other methods to check for consistency.

This guide provides a framework for tackling common problems in Unit 10. Remember to consult your specific homework assignment and course materials for the most accurate and relevant information. Good luck!