Questions: 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 250 speeding tickets (P0=250). Every year thereafter, the number of speeding tickets issued is predicted to grow by 5%. If Pn denotes the predicted number of speeding tickets during the year 2012+n, then Write the recursive formula for Pn Pn= × Pn-1 Write the explicit formula for Pn Pn= If this trend continues, how many speeding tickets are predicted to be issued in 2026? tickets

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 250 speeding tickets (P0=250). Every year thereafter, the number of speeding tickets issued is predicted to grow by 5%.

If Pn denotes the predicted number of speeding tickets during the year 2012+n, then Write the recursive formula for Pn
Pn= × Pn-1

Write the explicit formula for Pn
Pn=

If this trend continues, how many speeding tickets are predicted to be issued in 2026?
tickets

Transcript text: 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 250 speeding tickets $\left(P_{0}=250\right)$. Every year thereafter, the number of speeding tickets issued is predicted to grow by $5 \%$. If $P_{n}$ denotes the predicted number of speeding tickets during the year $2012+n$, then Write the recursive formula for $P_{n}$ $P_{n}=$ $\square$ $\times P_{n-1}$ Write the explicit formula for $P_{n}$ $P_{n}=$ $\square$ If this trend continues, how many speeding tickets are predicted to be issued in 2026? $\square$ tickets

Solution

Solution Steps

Step 1: Understanding the Problem

We are given an initial quantity $P_{0} = 250$, a rate of growth per period $r = 0.05$, and the number of periods $n = 14$. Our goal is to predict the future value $P_{n}$ based on an exponential growth model.

Step 2: Applying the Explicit Formula

The explicit formula to calculate the future value is: \[ P_{n} = P_{0} \cdot (1 + r)^{n} \] Substituting the given values into the formula, we get: \[ P_{n} = 250 \cdot (1 + 0.05)^{14} \]

Step 3: Calculating the Future Value

Using the formula, the calculated future value $P_{n}$ is approximately 495 when rounded to 0 decimal places.