Problem 27 - College Algebra

Question

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \(\$ 50 .\) Each month thereafter he increased the previous deposit amount by \(\mathrm{s} 20 .\) Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.

Step-by-Step Solution

Answer

Calculate monthly deposits and their cumulative sums, then graph the series over 12 months.

Step 1: Identify the Arithmetic Series

The problem describes a scenario where Javier's monthly deposits form an arithmetic series. The initial deposit (first term) is \(a = 50\), and each subsequent deposit increases by a constant amount, \(d = 20\). We need to calculate the sum of the deposits for each month to graph the cumulative series over one year.

Step 2: Calculate Monthly Deposits

The amount deposited each month follows the formula for the nth term of an arithmetic sequence: \(a_n = a + (n-1)d\). For each of the 12 months, calculate:- Month 1: \(a_1 = 50\)- Month 2: \(a_2 = 50 + 20 = 70\)- Month 3: \(a_3 = 50 + 40 = 90\)Continue this pattern to find the monthly deposits up to month 12.

Step 3: Determine Cumulative Deposits

Calculate the cumulative sum up to each month by adding all previous monthly deposits. For month \(n\), the cumulative sum \(S_n\) is the sum of all deposits from month 1 to month \(n\). You can use the formula for the sum of an arithmetic series: \(S_n = \frac{n}{2} (a_1 + a_n)\) or add manually.- Month 1: \(S_1 = 50\)- Month 2: \(S_2 = 50 + 70 = 120\)- Month 3: \(S_3 = 50 + 70 + 90 = 210\)Continue this until month 12.

Step 4: Graph the Series

Create a graph with the x-axis representing the months (1-12) and the y-axis representing the cumulative deposits. Plot each month as a point on the graph corresponding to its cumulative deposit, then connect these points with a line to show the growth pattern.

Key Concepts

Cumulative Deposits

Understanding how cumulative deposits work can be incredibly useful when managing savings. Instead of looking at deposits in isolation, cumulative deposits focus on the total amount of money accumulated over a period.

• At the start, Javier makes an initial deposit of \( \\(50 \).
• Every month, Javier adds more money, increasing his deposit by a fixed amount, \( \\)20 \), each time.
• This progressive addition means that each month, the total in the account grows, not just by the deposit for that month, but also by the sum of all previous deposits.

For every month up to the 12th, we sum all individual monthly deposits to get the total savings so far. For example, after the first month, he has \( \\(50 \); after the second month, the total becomes \( \\)120 \), because we add the second month's \( \\(70 \) to the initial \( \\)50 \).
Tracking cumulative deposits helps visualize saving patterns and encourages consistent savings practices.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This regularity is called the common difference.
In Javier’s case:

• The initial deposit is \(a = 50\).
• The common difference, representing the increase in deposit each month, is \(d = 20\).

The formula for finding the \(n\)-th term in an arithmetic sequence is \(a_n = a + (n-1)d\).Using this formula, you can calculate:

• The first month: \(a_1 = 50\).
• The second month: \(a_2 = 50 + 20 = 70\).
• Subsequent months follow this pattern, each adding \(20\) to the last month's deposit.

This predictable increase defines it as an arithmetic sequence, where you can forecast future values and understand how Javier’s savings grow month by month.

Graphing Arithmetic Series

Graphing an arithmetic series visually represents how cumulative values accumulate over time. This graphical approach makes it easier to perceive trends and growth patterns in data like savings.
For Javier’s savings:

• The x-axis represents the months, from 1 to 12.
• The y-axis represents the cumulative deposits made by Javier.

Each month’s cumulative deposit is plotted as a point on the graph. Here’s how you proceed:

• Month 1: Plot at \((1, 50)\).
• Month 2: Plot at \((2, 120)\).
• Continue similarly for all months up to month 12.

Once all points are plotted, you connect them with a line. This line illustrates the arithmetic growth of Javier's savings, showing a simple upward trend. The slope of this line reflects the regular monthly increase. Graphing helps make abstract numbers more tangible and is essential for understanding growth within an arithmetic series.