Problem 30 - Algebra 2 and Trigonometry

Question

On Monday, Elaine spent 45 minutes doing homework. On the remaining four days of the school week, she spent 15 minutes longer doing homework than she had the day before. a. Express the number of minutes Elaine spent doing homework on the \(n\) th day of the school week. b. Use sigma notation to represent the total number of minutes Elaine spent doing homework from Monday to Friday.

Step-by-Step Solution

Answer

Elaine's homework time for day \(n\) is \(15n + 30\) minutes. Total time from Monday to Friday is \(\sum_{n=1}^{5} (15n + 30)\).

Step 1: Determine Homework Duration for Each Day

Elaine spent 45 minutes on Monday doing homework. From Tuesday to Friday, she spent 15 minutes more than the previous day. Therefore, on Tuesday, she spent \(45 + 15 = 60\) minutes, on Wednesday \(60 + 15 = 75\) minutes, on Thursday \(75 + 15 = 90\) minutes, and on Friday \(90 + 15 = 105\) minutes.

Step 2: Express the Duration Algebraically

The time spent on each day forms an arithmetic sequence with the first term \(a_1 = 45\) and a common difference \(d = 15\). The general form of an arithmetic sequence for day \(n\) can be expressed as: \(a_n = a_1 + (n - 1) \times d\). Substituting the values, \(a_n = 45 + (n - 1) \times 15\). Simplifying gives: \(a_n = 15n + 30\).

Step 3: Represent Total Homework Time in Sigma Notation

The total amount of time spent from Monday to Friday is the sum of the sequence from day 1 to day 5. Using sigma notation, this can be expressed as: \[ \sum_{n=1}^{5} (15n + 30) \] This represents summing up the series \(15 \times 1 + 30, 15 \times 2 + 30, \ldots, 15 \times 5 + 30\).

Key Concepts

Sigma Notation

Sigma notation is a concise and efficient way to represent the sum of a set of numbers, often used with sequences and series. It allows us to express long sums without writing out every single term. This is especially useful in the case of arithmetic or geometric progressions where we can predict the sequence's behavior.
Sigma notation uses the Greek letter \(\Sigma\), which stands for summation. When using sigma notation, you describe:

• The starting index, indicating where the sum begins.
• The ending index, specifying where the sum stops.
• The general term of the sequence, which is the formula for the terms being added.

For Elaine's homework exercise, the problem requires you to express the total minutes spent from day 1 to day 5. This is denoted as:\[\sum_{n=1}^{5} (15n + 30)\]Here, "\(n\)" is the index that starts at 1 and ends at 5, which corresponds to the five school days. The expression \((15n + 30)\) calculates the homework time for each day. The sigma notation helps you quickly sum every day's homework time in a structured manner.

Arithmetic Progression

An arithmetic progression (or sequence) is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This constant is known as the common difference, often represented by \(d\).
This type of sequence is helpful in predicting future events and sums over a period, quite useful in Elaine's situation where her homework time increases uniformly.Elaine's homework scenario is a perfect example of an arithmetic sequence. She starts with 45 minutes on the first day (Monday), and each subsequent day she adds 15 more minutes than the previous day. This makes the common difference \(d = 15\).

• The first term \(a_1\) is 45 minutes, the time spent on Monday.
• The general term \(a_n\) for the \(n\)th day can be calculated using the formula: \(a_n = a_1 + (n-1) \times d\).

When stated, this becomes \(a_n = 45 + (n-1) \times 15\), simplifying to \(a_n = 15n + 30\). Each day's homework time can be found easily using this formula, making problem solving both quicker and more effective.

Sum of Sequences

Finding the sum of sequences, particularly arithmetic sequences, is a handy skill in mathematics. It allows you to determine the total of all terms within a specific sequence, without needing to calculate each term individually and then summing them up.
For an arithmetic sequence, the formula to calculate the sum, \(S_n\), of the first \(n\) terms is:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]Where:

• \(n\) is the number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the last term in the sequence.

In Elaine's scenario, this formula and the summation helps calculate the total minutes spent on homework over five days. The sigma notation \(\sum_{n=1}^{5} (15n + 30)\) simplifies the task of adding 45 for Monday, 60 for Tuesday, up to 105 for Friday. By separating the increasing homework time into an arithmetic sequence, users do not manually need to add each day’s amount of time, instead using the structured method offered by sigma notation and the sequence sum formula.