Question

Starting in the year 2012, the number of speeding tickets issued each year in Middletown is predicted to grow according to an exponential growth model. During the year 2012, Middletown issued 200 speeding tickets ( P 0 = 200 ). Every year thereafter, the number of speeding tickets issued is predicted to grow by 15%. If P n denotes the predicted number of speeding tickets during the year 2012 + n , then Write the recursive formula for P n P n = × P n − 1 Write the explicit formula for P n P n = If this trend continues, how many speeding tickets are predicted to be issued in 2025?

Answer 1

Answer:

Recursive formula:

$P_{n} = (1.15)P_{n-1}$

P(0) = 200

1070 speeding tickets are predicted to be issued in 2025

Step-by-step explanation:

The number of speeding tickets in an year, in function of the number in the previous year, can be given by the following formula:

$P_{n} = (1+r)P_{n-1}$

In which $P_{n-1}$ is the number of tickets issued in the previous year and r is the rate the number of tickets increase.

Every year thereafter, the number of speeding tickets issued is predicted to grow by 15%

So r = 0.15 and

$P_{n} = (1.15)P_{n-1}$

During the year 2012, Middletown issued 200 speeding

So P(0) = 200.

Then the recursive formula is:

$P_{n} = (1.15)P_{n-1}$

P(0) = 200

Write the explicit formula for P n P n = If this trend continues, how many speeding tickets are predicted to be issued in 2025?

2025 is 13 years after 2012.

So this is P(12).

We can expend the equation in the following format

$P(12) = (1.15)^{12}P(0) = (1.15)^{12}*200 = 1070$

1070 speeding tickets are predicted to be issued in 2025