Question

The data given below shows the production rate of a cotton factory. The first 1 ton is produced in the first five days, $\frac{1}{2}$ ton is produced in the next five days, and $\frac{1}{4}$ ton in the next five days. If the production rate continues in this pattern, how long would the factory take to produce 2 tons of cotton?

Answer 1

Following the pattern, we find that the production is halved in every five days .

Hence, the total production in the first 15 days would be:

$1+\frac{1}{2}+\frac{1}{4}\text{tons}$

Similarly, the total production after an inifinite time will be the sum of the infinite series:

$D=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$

Such an infinite series will eventually have very small terms as the denominator will keep growing.
So, even after a large number of terms, the sum will not always increase.

Subtracting 1 from both sides, we get:

$\Rightarrow D–1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$

Also, dividing the sum 'D' by 2, we get:

$\frac{D}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+...$

$\Rightarrow \frac{D}{2}=D–1$

$\Rightarrow \frac{D}{2}=1$

$\Rightarrow D=2tons$

Hence, the cotton production would be able to cover 2 tons in an infinitely long time.