To begin, we need to define how we’re going to measure [latex]n[/latex]. Remember that [latex]P_0[/latex] is the population when [latex]n = 0[/latex], so we probably don’t want to literally use the year [latex]0[/latex]. Since we already know the population in 2003, let us define [latex]n = 0[/latex] to be the year 2003.

Then [latex]P_0 = 12,000[/latex].

Next we need to find [latex]d[/latex]. Remember [latex]d[/latex] is the growth per time period, in this case growth per year. Between the two measurements, the population grew by [latex]15,000-12,000 = 3,000[/latex], but it took [latex]2007-2003 = 4[/latex] years to grow that much. To find the growth per year, we can divide: [latex]3000[/latex] elk /[latex]4[/latex] years =[latex]750[/latex] elk in [latex]1[/latex] year.

Alternatively, you can use the slope formula from algebra to determine the common difference, noting that the population is the output of the formula, and time is the input.

[latex]d=slope=\frac{\text{changeinoutput}}{\text{changeininput}}=\frac{15,000-12,000}{2007-2003}=\frac{3000}{4}=750[/latex]

We can now write our equation in whichever form is preferred.

Recursive form

[latex]P­₀ = 12,000[/latex]

[latex]P_n = P_{n-1} + 750[/latex]

Explicit form

[latex]P_n = 12,000 + 750_n[/latex]

To answer the question, we need to first note that the year 2014 will be [latex]n = 11[/latex], since 2014 is [latex]11[/latex] years after 2003. The explicit form will be easier to use for this calculation:

[latex]P_{11}= 12,000 + 750(11) = 20,250[/latex] elk

Plotting this data, it appears to have an approximately linear relationship:

While there are more advanced statistical techniques that can be used to find an equation to model the data, to get an idea of what is happening, we can find an equation by using two pieces of the data – perhaps the data from 1993 and 2003.

Letting [latex]n = 0[/latex] correspond with 1993 would give[latex]P_0 = 111[/latex] billion gallons.

To find[latex]d[/latex], we need to know how much the gas consumption increased each year, on average. From 1993 to 2003 the gas consumption increased from [latex]111[/latex] billion gallons to [latex]133[/latex] billion gallons, a total change of [latex]133 – 111 = 22[/latex] billion gallons, over [latex]10[/latex] years. This gives us an average change of [latex]22[/latex] billion gallons [latex]/10[/latex] year = 2.2[/latex] billion gallons per year.

Equivalently,

[latex]d=slope=\frac{\text{changeinoutput}}{\text{changeininput}}=\frac{133-111}{10-0}=\frac{22}{10}=2.2[/latex]billion gallons per year

We can now write our equation in whichever form is preferred.

Recursive form

[latex]P_0 = 111[/latex]

[latex]P_n = P_{n-1} + 2.2[/latex]

Explicit form

[latex]P_n = 111 + 2.2_n[/latex]

Calculating values using the explicit form and plotting them with the original data shows how well our model fits the data.

We can now use our model to make predictions about the future, assuming that the previous trend continues unchanged. To predict the gasoline consumption in 2016:

[latex]n = 23[/latex] (2016 – 1993 = 23 years later)

[latex]P_23 = 111 + 2.2(23) = 161.6[/latex]

Our model predicts that the US will consume [latex]161.6[/latex] billion gallons of gasoline in 2016 if the current trend continues.

To find when the consumption will reach [latex]200[/latex] billion gallons, we would set [latex]P_n = 200[latex], and solve for n:

[latex]P_n = 200[/latex]                                  Replace[latex]P_n[/latex] with our model

[latex]111 + 2.2_n = 200[/latex]                   Subtract [latex]111[/latex] from both sides

[latex]2.2_n = 89[/latex]                                Divide both sides by [latex]2.2[/latex]

[latex]n = 40.4545[/latex]

This tells us that consumption will reach [latex][/latex]200[latex][/latex] billion about 40 years after 1993, which would be in the year 2033.

The value for [latex]P_0[/latex] in this equation is [latex]70[/latex], so the initial starting cost is [latex]$70[/latex]. This tells us that there must be an initiation or start-up fee of [latex]$70[/latex] to join the gym.

The value for [latex]d[/latex] in the equation is [latex]30[/latex], so the cost increases by [latex]$30[/latex] each month. This tells us that the monthly membership fee for the gym is [latex]$30[/latex] a month.