Calculating Profits Using Arithmetic and Geometric Sequences

Understanding Sequences in Profit Calculations

When dealing with profit changes over time, two types of sequences commonly appear:

Arithmetic Sequence

• Profits increase or decrease by the same fixed amount each period.
• The common difference (D) is positive for increases and negative for decreases.
• Example phrases: "increased by $25", "decreased by $25".

Geometric Sequence

• Profits change by a fixed percentage of the previous period.
• The common ratio (r) is calculated as 1 plus or minus the percentage change.
• Example phrases: "increased by 10%", "decreased by 5%", "95% of the previous week".

Example Problems

1. Arithmetic Sequence: Profit Increasing by $25 Weekly

• Initial profit (U1): $400
• Common difference (D): +$25
• Find profit in the 5th week (U5):
  • Formula: U5 = U1 + (5 - 1) × D
  • Calculation: 400 + 4 × 25 = $500

2. Arithmetic Sequence: Profit Decreasing by $25 Weekly

• Initial profit (U1): $400
• Common difference (D): -$25
• Find profit in the 5th week (U5):
  • Formula: U5 = U1 + (5 - 1) × D
  • Calculation: 400 + 4 × (-25) = $300

3. Geometric Sequence: Profit Increasing by 10% Weekly

• Initial profit (U1): $400
• Common ratio (r): 1 + 10% = 1.1
• Find profit in the 5th week (U5):
  • Formula: U5 = U1 × r^(5 - 1)
  • Calculation: 400 × 1.1^4 ≈ $500.86
  • Rounded to nearest cent: $500.86

4. Geometric Sequence: Profit Decreasing by 5% Weekly

• Initial profit (U1): $400
• Common ratio (r): 1 - 5% = 0.95
• Find profit in the 5th week (U5):
  • Formula: U5 = U1 × r^(5 - 1)
  • Calculation: 400 × 0.95^4 ≈ $325.80

5. Geometric Sequence: Profit is 95% of Previous Week

• This is equivalent to a 5% decrease.
• Same calculation as above with r = 0.95.

Key Takeaways

• Identify whether the problem describes an arithmetic or geometric sequence based on wording.
• Use the appropriate formula:
  • Arithmetic: Un = U1 + (n - 1) × D
  • Geometric: Un = U1 × r^(n - 1)
• Convert percentage increases/decreases to decimal form for calculations.
• Round financial answers to the nearest cent for accuracy.

By applying these principles, you can accurately calculate profits over time in various business scenarios. For a deeper understanding of sequences, consider reading Mastering Sequence and Series: A Comprehensive Guide. If you're interested in the foundational concepts of arithmetic, check out Understanding Addition and Subtraction: Basics of Arithmetic. Additionally, for a broader perspective on ratios and proportions, see Understanding Averages, Ratios, and Proportions in Mathematics.