Problem 40 - Algebra 2 and Trigonometry

Question

Sean has started an exercise program. The first day he worked out for 30 minutes. Each day for the next six days, he increased his time by 5 minutes. a. Write the sequence for the number of minutes that Sean worked out for each of the seven days. b. Write a recursive definition for this sequence.

Step-by-Step Solution

Answer

The sequence is 30, 35, 40, 45, 50, 55, 60. Recursive definition: \( a_1 = 30 \), \( a_n = a_{n-1} + 5 \) for \( n \geq 2 \).

Step 1: Understanding the Problem

We know that Sean starts his exercise program with 30 minutes of workout and increases his workout time by 5 minutes each subsequent day for six days. We need to determine how long he works out each of these days, which forms a sequence.

Step 2: Determine the Sequence

The workout time for each day increases by 5 minutes. Thus, the sequence starts at 30 minutes on the first day. Each subsequent day's workout adds 5 minutes to the previous day's total. We calculate as follows: - Day 1: 30 minutes - Day 2: 30 + 5 = 35 minutes - Day 3: 35 + 5 = 40 minutes - Day 4: 40 + 5 = 45 minutes - Day 5: 45 + 5 = 50 minutes - Day 6: 50 + 5 = 55 minutes - Day 7: 55 + 5 = 60 minutes Thus, the sequence is: 30, 35, 40, 45, 50, 55, 60.

Step 3: Write the Recursive Definition

A recursive sequence is defined by a starting term and a rule that links each term to the previous one. Here, Sean's workout time can be described recursively as:- Base case (starting point): \( a_1 = 30 \)- Recursive rule: \( a_n = a_{n-1} + 5 \) for \( n \geq 2 \)This means the first day is 30 minutes and each following day Sean adds 5 minutes to the previous day's workout time.

Key Concepts

Recursive Sequences

A recursive sequence is a special type of sequence where each term is defined based on the previous term or terms. Unlike an arithmetic sequence, which uses a fixed formula for all terms, recursive sequences build on themselves incrementally.
For Sean's exercise routine:

• Starting Point (Base Case): The first day's workout, denoted as \( a_1 \), is 30 minutes.
• Recursive Rule: Each subsequent day's workout, \( a_n \), is determined by adding 5 minutes to the previous day's workout, \( a_{n-1} \). This can be described mathematically as: \( a_n = a_{n-1} + 5 \) for \( n \geq 2 \).

The beauty of recursive sequences lies in their simplicity and ability to express patterns through a single rule, building from a known point.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a specific pattern. These sequences are fundamental in mathematics as they help describe and predict behaviors within a given context.
In Sean's workout example, we have an arithmetic sequence, which is a type of mathematical sequence. Here:

• Each term after the first is obtained by adding a constant, which is called the common difference.
• For Sean's workout, the common difference is 5 minutes.
• The sequence starts with 30 and continues as: 30, 35, 40, 45, 50, 55, 60.

Mathematical sequences like these are powerful because they reveal consistent patterns and enable calculations and predictions about future values in the series.

Series and Patterns

Series and patterns are crucial concepts linked with sequences, providing insights into the sum of sequences and the repetition of elements within them.
A series is what you obtain when you sum all the terms of a sequence. In Sean's context, should you sum his workout times over the seven days, you'd get the total exercise time.

• The sum of Sean's sequence is: \(30 + 35 + 40 + 45 + 50 + 55 + 60\), totaling 315 minutes over seven days.

Understanding patterns is about recognizing the regularities or rules governing a sequence. With Sean's workout pattern, we recognize it by identifying the common difference of 5 minutes increase each day. Recognizing such patterns helps in predicting future terms and forming logical deductions.