Modelling with Sequences

Arithmetic models rely on Linear Growth, where the rate of change is constant. This is mathematically equivalent to a linear function $f(n) = dn + (a-d)$.

Geometric models rely on Exponential Growth or Decay, where the rate of change is proportional to the current value. This is equivalent to an exponential function $f(n) = a \cdot r^{n-1}$.

The choice between models depends on the nature of the increment: 'adding 50 dollars' implies an AP, whereas 'increasing by 5%' implies a GP.

• Step 1: Identify the Sequence Type: Look for keywords like 'fixed amount' (AP) or 'percentage/ratio' (GP).

• Step 2: Define Parameters: Determine the first term ($a$), the change factor ($d$ or $r$), and the target time ($n$).

• Step 3: Select the Objective: Decide if the problem asks for a specific value at a point in time ($u_n$) or a cumulative total ($S_n$).

• Step 4: Apply Formulas: Use $u_n = a + (n-1)d$ or $u_n = ar^{n-1}$ for specific terms, and the respective sum formulas for totals.

• Check the 'n' value: Always verify if the first term corresponds to 'Year 0' or 'Year 1'. If a value starts at $t=0$, the value after $k$ years is often the $(k+1)$-th term.

• Percentage Conversion: For a percentage increase of $p\%$, the common ratio is $r = 1 + \frac{p}{100}$. For a decrease, $r = 1 - \frac{p}{100}$.

• Total vs. Specific: If the question asks for 'total profit over 10 years', use the sum formula ($S_{10}$). If it asks for 'profit in the 10th year', use the term formula ($u_{10}$).

• Units and Rounding: In financial models, ensure your final answer is rounded to the appropriate currency units (e.g., 2 decimal places) unless specified otherwise.

• Confusing $d$ and $r$: Students often use an AP model for percentage changes because the percentage 'looks' like a constant number, but the actual value added changes every time.

• Off-by-one errors: Miscalculating $n$ is the most common error. For example, the number of years between 2020 and 2030 inclusive is 11, not 10.

• Incorrect $r$ for growth: Using $r = 0.05$ for a $5\%$ increase instead of $r = 1.05$. A ratio less than 1 always results in a decreasing sequence.