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Summation by Sigma Method

A snail start...

Question

A snail starts moving towards a point $3$ cm away at a pace of $1$ cm per hour. As it gets tired, it covers only half the distance compared to previous hour in each succeeding hour. In how much time will the snail reach his target?

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Solution

The snail movement can be thought of that of a G.P.
$1, 0.5, 0.25, . . . .$
$1 + 0.5 + 0.25 + . . . = 3$
$\frac{1 - {0.5}^{n}}{1 - 0.5} = 3$
$1 - {0.5}^{n} = 1.5$
${0.5}^{n} = - 0.5$
which is not possible.Hence,it never reaches its target.

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