Question

Create a table of points for a linear and an exponential function and share them below. Looking at the tables, imagine you are offered two different ways to receive an allowance. Option A increases by a common number (linear) every week, and Option B uses a constant multiplier (exponential) every week. If you plan to save money for a whole year, which would you prefer and why?2. Does your answer change if you only save for two weeks?

Answer 1

Here are two sample functions (same starting allowance, different growth types) and a table of points for several weeks.

- Linear (Option A): start $5 and add $2 each week
f(n) = 5 + 2n
- Exponential (Option B): start $5 and multiply by 1.5 each week
g(n) = 5 · (1.5)^n

Table of values (n = week):

Week | Linear f(n) = 5+2n | Exponential g(n)=5·(1.5)^n
0 | $5.00 | $5.00
1 | $7.00 | $7.50
2 | $9.00 | $11.25
4 | $13.00 | $25.31
8 | $21.00 | $128.14
12 | $29.00 | $648.73
26 | $57.00 | ~$189,379.08
52 | $109.00 | ~ $7,176,000,000 (≈7.18 billion)

Which would you prefer for a whole year (52 weeks)?
- Choose Option B (exponential). With a multiplier greater than 1, exponential growth compounds and quickly outpaces the linear increase. Even though both start the same, by a year the exponential option is astronomically larger.

Does the answer change if you only save for two weeks?
- In this example, no — after 1 week Option B is $7.50 vs $7.00, and after 2 weeks Option B is $11.25 vs $9.00, so Option B is still better.
- In general, it depends on the exact numbers: if the weekly multiplier is >1 (growth) and the exponential starts at the same or a similar amount, exponential usually becomes better quickly; if the multiplier is ≤1 or the linear “add” is very large relative to the multiplier, the linear option could be better for short times.