Question

Write an expression in sigma notation for each situation. The total weekly caloric intake for a diet that starts with 2000 calories on the first day and decreases by \(10 \%\) each day, so that on the \(i^{\text {th }}\) day it is \(2000(0.9)^{i-1}\).

Short answer

Question: Write a sigma notation expression for the total weekly caloric intake following a diet that begins at 2000 calories on day 1 and decreases by 10% daily. Answer: \(\sum^{7}_{i=1} 2000(0.9)^{i-1}\)

Step-by-step solution

Identify the variables of the sequence

In this case, the first day caloric intake (a_1) is 2000, and the common ratio (r) is 0.9, since the intake decreases by 10% (1-0.1) each day.

Express the general term (i-th term) of the sequence

We are given the general term expression of the sequence, which represents the caloric intake for the i-th day. We will use the index i to represent the days of the week from 1 to 7. The general term of the sequence, a_i, can be expressed as: a_i = 2000(0.9)^{i-1}

Write the sigma notation for the total weekly caloric intake

It is crucial to express the total weekly caloric intake as the summation of each day's intake. Sigma notation is employed to represent this summation: \(\sum^{7}_{i=1} 2000(0.9)^{i-1}\) In this case, we are summing the daily caloric intake from the 1st day (i=1) to the 7th day (i=7).

Key concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." In the problem at hand, the caloric intake of a diet decreases by 10% each day. This creates a geometric sequence where each day's caloric intake is 90% of the previous day's intake. The initial amount is 2000 calories, which serves as the first term of the sequence. Each subsequent term, or day's caloric intake, is computed by multiplying the previous day's calories by the common ratio of 0.9.

• The first term (\( a_1 \)) is 2000 calories.
• The common ratio (\( r \)) is 0.9.
• The formula for the \( i^{\text{th}} \) term is \( a_i = 2000(0.9)^{i-1} \).

The geometric sequence formula allows us to quickly determine how many calories are consumed each day and helps to track the overall intake over a period of time.

Caloric Intake

Caloric intake refers to the number of calories a person consumes through eating and drinking. This is essential for maintaining energy levels and supporting bodily functions. In this specific scenario, the caloric intake starts at 2000 calories on the first day. Each subsequent day, the intake decreases by \(10 \%\). This means the body receives less energy each day due to the geometric nature of decrement.

• Day 1: Consumption starts at 2000 calories.
• Every day afterward: The intake is multiplied by 0.9 (or, in other words, decreased by 10%).
• Day 2: \( 2000 \times 0.9 = 1800 \text{ calories} \)
• Day 3: \( 1800 \times 0.9 = 1620 \text{ calories} \)

This intentional reduction over the days may serve various dietary goals such as weight management or metabolic adjustments. It's important to calculate the intake correctly to ensure that dietary plans meet the individual's nutritional needs.

Summation

Summation is a powerful mathematical tool used to add up a sequence of numbers. Sigma notation, represented by the Greek letter sigma (\( \sum \)), is used to denote summation concisely. It effectively summarizes the total of a list of terms in a straight-forward way. In the dietary scenario, the weekly total caloric intake is expressed in sigma notation by summing the caloric intake from day 1 to day 7. In the given problem, the expression:\[\sum^{7}_{i=1} 2000(0.9)^{i-1}\]denotes that we are adding up the caloric intake for each day of the week. This allows us to find the week-long calorie consumption effectively.

• Top number (7): Indicates the final day (7th day) of the diet plan.
• Bottom number (1): Indicates the starting day of summation (1st day).
• Expression to the right of sigma: Shows the caloric intake formula used for each day.

This notation saves time while ensuring accuracy in calculating the total intake over the projected duration.