The Power of the Ratio
See how a single multiplier shapes the entire curve of a Geometric Progression.
Equation
y = 2 * 2^(x - 1)
Graph
Table
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
What this lesson covers
What you do
You shape the function y = a * r^(x - 1) and watch the graph answer.
Challenges to clear
- Set a = 2 and find r so the 4th term (x = 4) is 54 — three multiplications from the start.
- Same ratio: the 6th term explodes to 486. That is the power of the multiplier.
- Set the first term to 3.
- Now find the ratio that makes the 5th term 48. Then look at term 7 — that is the power of a multiplier.
Check yourself
In the formula t_n = a * r^(n-1), what is the specific role of 'a'?
- It determines how fast the sequence grows or shrinks.
- It is the first term of the progression. — correct
- It is the common ratio between consecutive terms.
- It shifts the entire graph horizontally.
Think about it
- If you increase r from 2 to 3, what happens to the 4th term?
- If you change a, does the shape of the curve change, or just its starting point?