Problem 42 - Algebra 2 and Trigonometry

Question

Julie is trying to lose weight. She now weighs 180 pounds. Every week for eight weeks, she was able to lose 2 pounds. a. List Julie's weight for each week. b. Write a recursive definition for this sequence.

Step-by-Step Solution

Answer

a. 180, 178, 176, 174, 172, 170, 168, 166, 164 b. \( w_n = w_{n-1} - 2 \), \( w_0 = 180 \)

Step 1: Understanding Julie's Initial Weight and Weekly Loss

Julie starts with a weight of 180 pounds. She loses 2 pounds each week. Therefore, each week, her weight will be reduced by 2 pounds from the previous week's weight.

Step 2: Calculate Weekly Weights

We will calculate Julie's weight at the end of each week by subtracting 2 pounds from the previous week's weight. Starting from week 0 (initial weight) to week 8, her weights will be: Week 0: 180 pounds, Week 1: 178 pounds, Week 2: 176 pounds, Week 3: 174 pounds, Week 4: 172 pounds, Week 5: 170 pounds, Week 6: 168 pounds, Week 7: 166 pounds, Week 8: 164 pounds.

Step 3: Writing the Recursive Definition

A recursive definition needs two components: an initial condition and a recursive equation. The initial weight is given as 180 pounds at week 0. The recursive formula for the weight sequence can be expressed as: \[ w_n = w_{n-1} - 2 \]where \( w_n \) is the weight at week \( n \) and \( w_0 = 180 \). This formula captures the weekly loss of 2 pounds starting from the initial weight.

Key Concepts

Weight Loss Sequence

A weight loss sequence, like the one Julie is experiencing, is simply a list of her weights over time as she loses a consistent number of pounds each week.
This kind of sequence helps track progress and maintain focus on your goal. In Julie's case, she starts at 180 pounds and loses 2 pounds every week.
Listing her weights week by week allows us to see this sequence in action:

• Week 0: 180 pounds
• Week 1: 178 pounds
• Week 2: 176 pounds
• Week 3: 174 pounds
• Week 4: 172 pounds
• Week 5: 170 pounds
• Week 6: 168 pounds
• Week 7: 166 pounds
• Week 8: 164 pounds

By plotting these numbers, you create a sequence that visually illustrates her consistent, week-by-week weight loss. This type of pattern is common in both personal and scientific contexts, as it enables easy monitoring of continual changes.

Initial Condition

The initial condition in a sequence sets the stage for the entire process. It is the starting point from which all calculations are made.
For Julie, the initial condition is her weight at week 0, which was 180 pounds.
This initial value is crucial because it determines the first point in the sequence. From this point, subsequent elements in the sequence are derived using a predetermined process.
Without a clear initial condition, it's impossible to correctly track or predict future values in a sequence. Always define the initial condition as precisely as possible to maintain accuracy in your sequence.

Recursive Equation

In mathematics, a recursive equation is a formula that defines each term of the sequence using the previous terms.
It's a way of expressing changes without listing all possible outcomes up front.
For Julie’s weight loss sequence, the recursive equation is:\[ w_n = w_{n-1} - 2 \]Here, \( w_n \) represents her weight in week \( n \) and \( w_{n-1} \) is her weight from the previous week. The "-2" signifies her weekly weight loss of 2 pounds.
Recursive equations save time and reduce complexity by focusing only on changes occurring from one term to the next. They're powerful tools for sequences involving predictable patterns. If you're dealing with repeating changes, try representing it through a recursive formula. It's an efficient method to clearly convey sequential developments.