Problem 11 - Algebra for College Students

Question

Sue is saving quarters. She saves 1 quarter the first day, 2 quarters the second day, 3 quarters the third day, and so on for 30 days. How much money will she have saved in 30 days? \(\$ 116.25\)

Step-by-Step Solution

Answer

Sue will have saved $116.25.

Step 1: Identify Pattern

The problem describes Sue saving quarters by incrementing the number of quarters saved each day by one additional quarter. This forms an arithmetic sequence where the first term (a_1) is 1 quarter, the second term (a_2) is 2 quarters, up to the 30th term (a_30) which is 30 quarters.

Step 2: Determine Sequence Sum Formula

To find the total quarters saved, we need to sum the arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is given by \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last (nth) term.

Step 3: Calculate Total Quarters

Substitute the known values into the formula: \( n = 30 \), \( a_1 = 1 \), and \( a_n = 30 \). Thus, the total number of quarters savings is\[ S_{30} = \frac{30}{2} \times (1 + 30) = 15 \times 31 = 465 \] quarters.

Step 4: Convert Quarters to Dollars

Since 1 quarter equals \( \\(0.25 \), to find the total dollar amount, multiply the total quarters by 0.25:\[ \text{Total savings in dollars} = 465 \times 0.25 = \\)116.25 \].

Key Concepts

Sum of Arithmetic Series

Understanding how to find the sum of an arithmetic series is important in many mathematical applications. In this exercise, Sue’s savings pattern over 30 days forms an arithmetic sequence, where each day the number of quarters saved increases by one. To find the total number of quarters saved, we use the formula for the sum of an arithmetic series.The formula for the sum of the first \(n\) terms of an arithmetic sequence is \( S_n = \frac{n}{2} \times (a_1 + a_n) \). This formula allows us to add up terms efficiently instead of adding each term individually.

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

Once you substitute the known values—\( n = 30 \), \( a_1 = 1 \), and \( a_{30} = 30 \)—into the formula, you can calculate the total number of quarters saved as \( S_{30} = \frac{30}{2} \times (1 + 30) = 15 \times 31 = 465 \) quarters. It's efficient and saves time, especially when dealing with larger sequences.

Sequence Pattern Recognition

Recognizing patterns in sequences is a key skill in mathematics. It helps in understanding and predicting future elements. In this exercise, the sequence is based on Sue increasing the number of quarters she saves each day by one.For arithmetic sequences, each term increases by a constant amount, called the common difference. In Sue's case, the common difference is 1, because each day she saves one more quarter than the previous day.

• The first term \( a_1 \) is 1 quarter saved on the first day.
• Each following term increases by 1 additional quarter.

Recognizing this pattern allows us to set up the arithmetic sequence formula where any term \( a_n \) can be expressed as \( a_n = a_1 + (n-1) \ imes d \). Understanding this pattern helps you apply the right formula to solve these problems effectively.

Savings Calculation

The ultimate goal in this problem is to find out the total amount of money saved by the end of 30 days. Once Sue's total quarters saved are calculated as 465 using the sum of arithmetic series, converting this figure into dollars is the final step.One quarter is equal to \(0.25. Therefore, to find the total savings in dollars, you multiply the total number of quarters by 0.25.

• Total quarters: 465
• Conversion rate: 1 quarter = \)0.25

The total savings in dollars is \( 465 \times 0.25 = 116.25 \) dollars. This conversion is simple but crucial to understanding how savings can accumulate, illustrating the power of consistent saving over time. Effective savings calculation involves recognizing the pattern of savings, applying the right formulas, and converting your findings into usable denominations like dollars.