IM Commentary
The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.
Solutions
Solution:
Table
a. A table of values giving the number of bags of leaves and the amount paid
using methods 1 and 2 shows that method 1 pays more up to and including
eleven bags.
b. The table also shows that method 2 pays more as soon as Celia rakes at least
twelve bags of leaves. We know that method 2 will always pay more, beyond
the twelfth bag, because doubling an amount $x$ gives a larger increase
than adding 2 as soon as $x$ is greater than 2:
$$
2x > x + 2\quad\text{whenever}\quad x > 2.
$$
| Number of Bags |
Payment Method 1 (dollars) |
Payment
Method 2 (dollars) |
| 1 |
2 |
0.02 |
| 2 |
4 |
0.04 |
| 3 |
6 |
0.08 |
| 4 |
8 |
0.16 |
| 5 |
10 |
0.32 |
| 6 |
12 |
0.64 |
| 7 |
14 |
1.28 |
| 8 |
16 |
2.56 |
| 9 |
18 |
5.12 |
| 10 |
20 |
10.24 |
| 11 |
22 |
20.48 |
| 12 |
24 |
40.96 |
Solution:
2. arithmetic and geometric sequences
The numbers in the second column of the table in the first solution form part of the arithmetic
sequence which starts with 2 and increases each time by 2: the $n^{th}$ term
in this arithmetic sequence is $2n$ and this is the number in the $n^{th}$ row
of the second column of the table. The third column of the table is a geometric
sequence which starts at $0.02$ and increases by multiples of 2 each
time. The $n^{th}$ term in this sequence, found in the $n^{th}$ row of
the third column, is
$$
\frac{2^n}{100}.
$$
The numerator $2^n$ shows the geometric sequence, while the denominator
$100$ reflects the fact that the sequence began at $\frac{2}{100}$.
Geometric sequences grow exponentially. Since the multiplier two is larger than
one, the geometric sequence grows faster than, and eventually surpasses, the linear arithmetic sequence. To see this more clearly, note that each additional bag of leaves makes Celia two dollars with method 1 while with method 2 it doubles her payment. Hence as soon as payment method 2 is worth more than two dollars (that is after 8 bags of leaves) method 2 pays more than method 1 for every additional bag and so
the deficit is quickly made up.
