During the winter, the amount of water that flows down a river remains at a low constant. In the spring, when the snow melts, th
e amount of water increases drastically, until it decreases to a steady rate in the summer. The flow then slowly decreases through the fall into the winter. Consider the graphs shown. Which graph best represents the given situation?
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Since are in arithmetic progression,
and since are in geometric progression,
Recall that
It follows that
so the left side is
Also recall that
so that the right side is
Solve for .
Now, the numerator increases more slowly than the denominator, since
and for ,
This means we only need to check if the claim is true for any .
doesn't work, since that makes
.
If , then
If , then
If , then
There is only one value for which the claim is true, .