Question

Suppose that five people are ill during the first week of an epidemic and each sick person spreads the contagious disease to four other people by the end of the second week and so on. By the end of ${15}^{th}$ week, how many people will be affected by the epidemic?

Answer 1

It is given that five people are ill during the first week and each sick person spreads the contagious disease to four other people by the end of the second week. This is going to continue in the after weeks. Let us write this as sequence as follows:

$5,5\left(4\right),5(4{)}^2,.....$ or $5,20,80,....$

We need to make this sequence as series because we have to find total number of people affected by the epidemic so the above sequence will become,

$5+20+80+.....$

In the above series, the first term is $a_1=5$, the second term is $a_2=20$ and so on.

We find the common ratio $r$ by dividing the second term by first term as shown below:

$r=\frac{20}{5}=4$

We know that the sum of an geometric series with first term $a$ and common ratio $r$ is $S_n=\frac{a(r^n-1)}{r-1}$ if $r>1$

Now, substitute $a=5,r=4$ and $n=15$ in $S_n=\frac{a(r^n-1)}{r-1}$ as follows:

$S_{15}=\frac{5[{\left(4\right)}^{15}-1]}{4-1}=\frac{5[{\left(4\right)}^{15}-1]}{3}=\frac{5}{3}[{\left(4\right)}^{15}-1]$

Hence the sum is $\frac{5}{3}[{\left(4\right)}^{15}-1]$.