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Snail With Confidence

A snail crawls along a seven inch-long rubber band.

After each minute, the rubber band is uniformly stretched (instantaneously!) by an additional one inch.

During the "stretch," the snail stays in place but is moved proportionally, i.e. if at one-third length of the rubber band, after the stretch it is still at one-third length of what is now a one-inch longer rubber band.

If it travels one inch per minute, will the snail ever reach the end of the rubber band?

 

Source: C. Kosniowski's Fun Mathematics on Your Microcomputer, 1983, p. 17


Hint: Set up a table showing times, length of rubber band, and current position of snail. See a pattern?

 


Solution Commentary: By setting up the table, you should see that at the time of N minutes, the snail has crawled N inches and the length of the rubber band is 7N inches.

The relative position of the snail to the rubber band's length is 1/7 + 1/14 + 1/21 + ... + 1/7N after N minutes.

This can be factored to produce (1/7)(1 + 1/2 + 1/3 + ... + 1/N), suggesting the Harmonic Series is involved....which is divergent and we thus know the relative position of the snail eventually exceeds 1.

That is, when N is about e7/(1.781) the series expression is 7/7, or the snail reaches the end of the rubber band!

At this stage, the time N required is slightly less than 10 1/4 hours.

Now, if the rubber band had been 10 inches long at the start, the snail would need more than 8 days to reach the end.

If the rubber band had been 100,000 inches long (slightly longer than 1.5 miles) at the start, the snail would reach the end....but would need a time length greater than the estimated age of the universe....so is it possible?

Ah, the power of the Harmonic Series strikes again!