At a certain time each year, the scaling down of weekly production of sun hats starts.Heating a large building requires $50000$ gallons of oil in $2013.$ As the building is extended, two

forecasts for growth in demand for oil are made:

Forecast $1$: increases of $5000$ gallons per year.

Forecast $2$: increases of $6\%$ each year.

Use series to calculate for each forecast:

$($a$)$ The number of gallons required in $2015,2020.$

$($b$)$ The total consumption of oil from $2015$ to $2020$ inclusive.

$($a$)$ If weekly production is $190000$ hats, and is reduced by $15200$ hats each week, use series to

calculate:

$($i$)$ The weekly output after $10$ weeks of scaling down.

$($ii$)$ The total production during that $10$ weeks.

$($b$)$ Calculate the weekly reduction if the production should be scaled down from $190000$ to

zero in $12$ weeks.