Losing weight

A 200 pound man starts a weight loss regimen that promises a loss of 1 percent of body weight per week.

How many weeks will it take to reach his goal of 150 pounds?

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The weight loss program has diminishing effectiveness each week.

In the first week, he will lose 200(0.01) = 2 pounds so he will weigh in at 198 pounds. In the second, he will lose a bit less weight: only 198(0.01) = 1.98 pounds. Each successive week he will weigh in lighter and therefore lose a bit less weight.

The easiest way to solve this problem is to track his weekly weight. If he loses 1 percent per week; that means each week he will weigh in at 99 percent of his previous week’s weight.

In other words:

W_t = 0.99W_t-1

We know that his initial weight is 200, so the formula for weight after n weeks is:

W_n = 0.99W_n-1 = (0.99)² W_n-2 = … = 200(0.99)ⁿ

We solve this equation for the target of 150 pounds to find:

150 = 200(0.99)ⁿ => n = ^(log150/200)/_log0.99 = 28.62…

 

Therefore, he will drop below 150 pounds at the 29th week. Of course few people have the discipline to stay on a diet for over half a year, and that’s perhaps one of the reasons weight loss is hard.