Learn how constant ratios create exponential patterns, model compound growth and decay, and unlock powerful tools for analyzing repeated percentage changes.
Our interactive approach pairs numerical tables with ratio reasoning, adding financial contexts and letting learners explore the ratio themselves through interactive controls.
Divide consecutive terms, look at the curve, and decide if you are modelling growth or decay. Our interactive explorer adds immediate visual cues to help you spot patterns.
Formula
a_n = 2.00 · 1.50n-1
Highlighted Term
a4 = 6.75
Behavior
Growth
What to notice
The Grade 12 book emphasises how repeated multiplication stacks factors. The Multiplier Lab below shows how the \(n-1\) exponent tracks those stacked multipliers in real time.
a1 = 4
r = 1.25
n = 6
nth term
a6 = 12.21
Computed with $a_n = a_1 r^{n-1}$
Partial sum
S6 = 45.04
Geometric series formula
| Term | Value | Multiplier explanation |
|---|---|---|
| a1 | 4.00 | Starting value |
| a2 | 5.00 | 1.25^{1} |
| a3 | 6.25 | 1.25^{2} |
| a4 | 7.81 | 1.25^{3} |
| a5 | 9.77 | 1.25^{4} |
| a6 | 12.21 | 1.25^{5} |
Understanding the pattern
Every geometric term is the previous term multiplied by the constant ratio. The sliders above help you see how the exponent \(n-1\) literally counts how many multipliers were applied.
Knowing the nth term lets you sum geometric series quickly. Slide the controls and narrate why the series formula changes when $$r = 1$$ versus $$r \ne 1$$.
$S_n = a_1 rac{r^n - 1}{r - 1}$ is derived by multiplying the series by \(r\), subtracting, and factoring. The interactive table shows those aligned rows.
Whether you are looking at compound interest or successively shrinking medication doses, the same multiplicative model applies. Adjust the sliders to test "what if" questions before you commit to a plan.
Start = 1200
Change = 8% → r = 1.080
Periods = 10
Focus = period 5
Focus period amount
1633
Total accumulated
17384
Series formula
Sn = a1 $\frac{r^n - 1}{r - 1}$
| Period | Amount | Multiplier |
|---|---|---|
| 1 | 1200 | starting value |
| 2 | 1296 | 1.080^{1} |
| 3 | 1400 | 1.080^{2} |
| 4 | 1512 | 1.080^{3} |
| 5 | 1633 | 1.080^{4} |
| 6 | 1763 | 1.080^{5} |
| 7 | 1904 | 1.080^{6} |
| 8 | 2057 | 1.080^{7} |
| 9 | 2221 | 1.080^{8} |
| 10 | 2399 | 1.080^{9} |
Real-world insight
In compound growth scenarios, the amount added each period is not constant, but the percentage is. Small ratio errors snowball over time, so visualizing them helps you reason about sustainable plans.
Use the tabs to progress from ratio basics to reasoning with sums and logarithms.
Can you identify ratios, compute terms, and reason with sums without a worked example next to you?