Question

The number of cases of a disease

increases by the same factor each year, as
shown in the table below.
Write an expression for the number of
cases of the disease after n years.
Start
End of year 1
End of year 2
End of year 3
Number of cases
1400
2100
3150
4725

Answer 1

The number of disease cases increases by a factor of 1.5 each year. The expression for the number of cases after n years is 1400 × 1.5n, with 'n' representing the number of years.

Step-by-step explanation:

The number of cases of a disease is increasing each year by the same factor. To find the expression for the number of cases after n years, we examine the pattern of increase from the table provided. The pattern suggests that there is an annual multiplicative factor, which we can identify as follows:

• Year 1: 1400 cases
• Year 2: 2100 cases (2100 / 1400 = 1.5)
• Year 3: 3150 cases (3150 / 2100 = 1.5)
• Year 4: 4725 cases (4725 / 3150 = 1.5)

Since the number of cases is increasing by a factor of 1.5 each year, we can define the initial number of cases as P, the growth factor as r, and the number of years as n. The expression for the number of cases after n years would then be:

P × rn

Substituting the values from the table, we get:

1400 × 1.5n

This formula gives us the projected number of cases after n years, assuming a consistent growth rate of 1.5 per year.

Answer 2

N*r^n

Step-by-step explanation:

Let the initial number of cases at the start of year 1 be represented by N.

From the given information, we know that the number of cases increases by the same factor each year. Let this factor be represented by r.

Then, at the end of year 1, the number of cases would be N*r, since it has increased by a factor of r.

Similarly, at the end of year 2, the number of cases would be Nrr, or N*r^2.

At the end of year 3, the number of cases would be Nrrr, or Nr^3.

We can use this pattern to write a general expression for the number of cases after n years:

N * r^n

where N is the initial number of cases, r is the common factor by which the number of cases increases each year, and n is the number of years elapsed.