Assume the sequence of annual retreats is quadratic: aₙ = an² + bn + c - Deep Underground Poetry

Understanding the Context

A quadratic sequence is defined by a second-degree polynomial where each term aₙ depends quadratically on its position n. The general form aₙ = an² + bn + c captures how values evolve when growth or decrease accelerates or decelerates over time.

In the context of annual retreats, this means the change in value from year to year isn’t constant—it increases or decreases quadratically, reflecting real-world dynamics such as climate change, erosion, or compounding financial losses.

Why Use the Quadratic Model for Annual Retreats?

Real-life phenomena like snowmelt, riverbed erosion, or funding drawdowns rarely grow or shrink steadily. Instead, the rate of change often accelerates:

Key Insights

• Environmental science: Snowpack decrease often quickens in warmer years due to faster melting.
  
• Finance: Annual reserves loss may grow faster as debts or withdrawals accumulate.
  
• Engineering: Wear and tear on structures can accelerate over time.

A quadratic model captures these nonlinear changes with precision. Unlike linear models that assume constant annual change, quadratic modeling accounts for varying acceleration or deceleration patterns.

How to Determine Coefficients a, b, and c

To apply aₙ = an² + bn + c, you need three known terms from the sequence—typically from consecutive annual retreat values.

Step 1: Gather data — Select years (e.g., n = 1, 2, 3) and corresponding retreat volumes a₁, a₂, a₃.

Final Thoughts

Step 2: Set up equations

Using n = 1, 2, 3, construct:

• For n = 1: a(1)² + b(1) + c = a + b + c = a₁
  
• For n = 2: a(2)² + b(2) + c = 4a + 2b + c = a₂
  
• For n = 3: a(3)² + b(3) + c = 9a + 3b + c = a₃

Step 3: Solve the system

Subtract equations to eliminate variables:

Subtract (1st from 2nd):
(4a + 2b + c) – (a + b + c) = a₂ – a₁ ⇒ 3a + b = a₂ – a₁