Arithmetic and Geometric Progressions

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 Arithmetic and Geometric Progressions are mathematical sequences that model linear and exponential changes, respectively. In business, Arithmetic Progressions map steady, flat-rate increases or decreases over time, whereas Geometric Progressions are vital for calculating compound interest, investment growth, and scaling trends. [1, 2, 3, 4]


1. Arithmetic Progression (AP)

An AP is a sequence of numbers where the difference between any two consecutive terms is always constant. This constant is called the common difference ($d$).
  • General Formula: $a_n = a + (n-1)d$
  • Sum Formula: $S_n = \frac{n}{2}[2a + (n-1)d]$
  • Where: $a = \text{first term}$, $n = \text{number of terms}$, $a_n = n^{th} \text{term}$.


Applications in Business
  • Straight-Line Depreciation: If a company purchases a delivery truck for $\$50,000$ and its value drops by $\$5,000$ annually, the salvage value follows an AP ($\$45,000, \$40,000, \$35,000$).
  • Fixed-Rate Loan Amortization: When paying off the principal of a loan in steady, equal increments over a set period (like a $\$1,000$ reduction per month).
  • Workforce Planning: Calculating total labor hours or salary costs if you add a fixed number of new employees each quarter.
2. Geometric Progression (GP)

A GP is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio ($r$).
  • General Formula: $a_n = a \cdot r^{n-1}$
  • Sum Formula: $S_n = \frac{a(1 - r^n)}{1 - r}$
  • Where: $a = \text{first term}$, $n = \text{number of terms}$, $r = \text{common ratio}$.
Applications in Business
  • Compound Interest & Investments: When calculating the future value of an investment or retirement portfolio that grows at an annual compounded rate.
  • Market Sizing & Growth: Forecasting future user bases, sales revenue, or market shares that scale exponentially at a fixed percentage (e.g., $10\%$ month-over-month growth).
  • Exponential Depreciation: Calculating the depreciated value of assets using the declining balance method, where the asset loses a specific percentage of its current value each year. [10]
Real-World Example

Suppose you forecast your business's social media advertising reach.
  • If reach grows by a steady 500 people per month, it follows an AP ($500, 1000, 1500$).
  • If your reach doubles by 2 \times every month, it follows a GP ($500, 1000, 2000$).



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