Question

Population of a City A city was incorporated in 2004 with a population of 35,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by

$P_n=35,000{\left(1.02\right)}^n$

(a) Find the first five terms of the sequence.

(b) Find the population in 2014.

Short answer

1. The first five terms are35700, 36414, 37142, 37885, 38643.
2. the population is 42665.

Step-by-step solution

a.Step 1.  Given

A city was incorporated in 2004 with a population of 35,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by

$P_n=35,000{\left(1.02\right)}^n$.

Step 2.  To determine

Find the first five terms of the sequence.

Step 3.  Calculation

To find the first five terms we will plug $n=1,2,3,4,5$respectively in the given function.

$\begin{array}{l}{P}_1=35000{\left(1.02\right)}^1=35700\\ P_2=35000{\left(1.02\right)}^2=36414\\ P_3=35000{\left(1.02\right)}^3=37142.28\\ P_4=35000{\left(1.02\right)}^4=37885.13\\ P_5=35000{\left(1.02\right)}^5=38642.83\end{array}$

So, the first five terms are35700, 36414, 37142, 37885, 38643.

b.Step 1.  Given

A city was incorporated in 2004 with a population of 35,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by

$P_n=35,000{\left(1.02\right)}^n.$

Step 2.  To determine

Find the population in 2014.

Step 3.  Calculation

In 2014, $n=2014-2004=10$.

So, the population is:

$P_{10}=35000{\left(1.02\right)}^{10}=42665$

Hence, the population is 42665.