Question

A school attendance clerk noticed that the first week after winter break 5 students were out with the flu. By the end of the second week, a total of 7 students had come down with the flu at one time and by the third week 10 had the flu at one time?

A school attendance clerk noticed that the first week after winter break 5 students were out with the flu. By the end of the second week, a total of 7 students had come down with the flu at one time and by the third week 10 had the flu at one time. The attendance clerk noticed the number of students that had the flu was growing exponentially with a growth factor of 1.4.

Write a recursive rule & a explicit rule:

If the model continues, how many would have the flu by the 12th week of school?

Answer 1

The recursive rule is `y_n = 1.4xxy_(n–1),` where `y_1=5`.
The explicit rule is `y = 5xx(1.4)^(x-1)`.
After `x=12` weeks, `y=202` students would have the flu.

Explanation:

A recursive rule is one where each value in a sequence is a function of the values before it.

Let `y_n` be the number of students with the flu after `n` weeks.
Since the growth rate in our model is exponential with a factor of 1.4, that means each week there are 1.4 times as many students with the flu as there were the previous week:

`y_2=1.4y_1`
`y_3=1.4y_2`
`y_4=1.4y_3`

and so on. This is written in a general form as

`y_n=1.4y_(n–1)`

This is useful if we know one value in the sequence and want to compute the next value (or values), but it's not practical for calculating an arbitrary term (like `y_12`).

An explicit rule is a function that relates a varying input value (like `x`) to a responding output value (like `y`). For a given value of `x`, we can plug it into the explicit rule and get a unique value for `y`.

Given the recursive rule above, we can derive the explicit rule as follows:

`y_2=1.4y_1 = 1.4xx(5)color(white)(xx 1.0 xx 1.0)=5xx(1.4)^1`
`y_3=1.4y_2 = 1.4 xx (1.4 xx 5)color(white)(xx1.0)=5xx(1.4)^2`
`y_4=1.4y_3 = 1.4 xx (1.4 xx 1.4 xx 5)= 5 xx (1.4)^3`

From here, it is not hard to see that after `n` weeks, the formula will be

`y_n = 5 xx (1.4)^(n-1)`

This can be written with `x` instead of `n` (since `x` is our usual input variable) and thus, without need of the `n` subscript:

`y = 5 xx (1.4)^(x-1)`

We now have `y` as a function of `x`. The value of `y` is the number of students with the flu after `x` weeks.

We can now use this explicit model to calculate the number of students with the flu after 12 weeks:

`y=5 xx (1.4)^(12–1)`
`color(white)(y)=5xx(1.4)^11`
`color(white)(y)=5xx40.49565...`
`color(white)(y)=202.478...`
`color(white)(y)~~202`

So, according to the exponential model given, there will be 202 students with the flu after 12 weeks.