Problem 45 - Algebra 2

Question

Writing Suppose you are to receive an allowance each week for the next 26 weeks. Would you rather receive (a) \(\$ 1000\) per week or (b) 2\(c\) the first week, 4\(c\) the second week, 8\(c\) the third week, and so on for the 26 weeks?

Step-by-Step Solution

Answer

The comparison between the two total allowances will reveal which of the two scenarios results in a higher total allowance over the given period.

Step 1: Calculate the total allowance for the first scenario

Since the allowance is constant at $1000 per week over 26 weeks, the total allowance for 26 weeks is simple multiplication: \(26 * \$1000 = \$26000.\)

Step 2: Calculate the total allowance for the second scenario

In the second scenario, the weekly allowance doubles every week for 26 weeks, starting from 2c. This is a geometric sequence where the common ratio \(r=2\), the first term \(a=2c=\$0.02\), and the number of terms \(n=26\). The sum of a geometric sequence can be calculated using the formula: \[S = a * \frac{1 - r^n}{1 - r}\] Substituting the given values into the formula gives the total allowance for the second scenario.

Step 3: Compare the total allowances from the two scenarios

After calculating the total allowances for the two scenarios, compare the results to determine which scenario results in a higher total allowance over the 26 weeks.

Key Concepts

Allowance Calculation

When calculating an allowance over a set period, it's essential to understand the terms involved. In this exercise, we consider two different methods of receiving an allowance over 26 weeks.

• In the first scenario, you receive a fixed amount of $1000 each week. Calculating the total for this scenario is straightforward: multiply the weekly allowance by the number of weeks.
  
  For example, $1000 per week for 26 weeks totals to $26,000.


• In the second scenario, the allowance doubles every week, starting from an initial amount 2c. This is a bit more complex as it involves a concept known as a geometric sequence.

Understanding allowance calculations like these helps in making informed financial decisions, as seen in comparing steady earnings versus exponential growth.

Geometric Series

A geometric series is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. In this exercise, the sequence of allowances is described by: 2c, 4c, 8c, and so forth.

• The first term: This is where the sequence starts. Here, it's 2c.


• The common ratio: This is the factor we multiply each term by to get the next term. The common ratio in this case is 2, as each week the allowance doubles.


• The number of terms: This is how many terms are in the sequence. For our exercise, there are 26 terms (representing weeks).

The sum of a geometric series can be found using the formula: \[S = a \times \frac{1 - r^n}{1 - r}\]where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Understanding geometric series prepares you to handle scenarios where growth isn’t linear.

Financial Comparison

When comparing financial options, it's crucial to analyze the total amounts received in each scenario. In this exercise, comparing the fixed weekly allowance with a growing allowance over 26 weeks demonstrates this well.

• Scenario A gives you a steady $26,000 over 26 weeks.


• Scenario B shows a potential exponential growth in allowance. By calculating the sum of this geometric sequence, you determine if it's higher or lower than the $26,000 from scenario A.

Financial comparison requires evaluating both options and their implications. A steady income might offer security, while an exponentially increasing one could pose higher risks and rewards. It's crucial to consider personal risk tolerance and financial goals when making such decisions.

Sequence Multiplication

Sequence multiplication involves applying a consistent multiplication factor throughout a series of numbers. In our allowance example, this is observed in scenario B, where the weekly allowance keeps getting multiplied by a factor of 2 each week, forming a geometric sequence.

• Start with a base amount: Here, 2c is the base that gets multiplied.


• Apply a consistent factor: Every subsequent term is obtained by multiplying the previous term by 2.

This type of sequence multiplication demonstrates how a seemingly small start value can rapidly increase to significant sums. In practical terms, it highlights the power of exponential growth, often seen in investments and savings where interests or investments compound over time.