Problem $1$

In the past, Peter Kelle's car dealership in Baton Rouge sold an average of $800$ Ford F$150\text{'}$s each year. In the past two years $200$ and $250,$ respectively, were sold in the fall; $250$ and $200$ were sold in winter; $50$ and $65$ were sold in the spring; and $300$ and $285$ were sold in the summer.

What are the seasonal indices for each season?

Fall: $\frac{200+250}{2}=\frac{225}{800}=0\mathrm{.}28$

Winter: $\frac{250+200}{2}=\frac{225}{800}=0\mathrm{.}28$

Spring: $\frac{50+65}{2}=\frac{55}{800}=0\mathrm{.}06$

Summer: $\frac{300+285}{2}=\frac{250}{800}=0\mathrm{.}32$

$\frac{800}{4}=200$

Peter is planning a major expansion in his dealership as a local minor competitor has just closed. He expects demand for Ford F$150\text{'}$s to increase according to the following trend line:

Demand $=800+50T$

where $T$ is the number of years from today $($i$.$e$.$ this year is year zero, next year is year one$)$

Using the above seasonal indices and trend line, predict Peter's sales during fall, winter, spring, and summer $8$ years from now.

Fall: $800+50(8)=1,100$

Summer: $300+50(8)=1,100$

Winter: $250+50(8)=1,150$

Spring: $50+50(8)=1,050$