You are considering buying a refrigerator. A new model would lower your electricity bills from $$1200$ per year to $$800$ per year, because your current refrigerator is very inefficient. The refrigerator you want is sold at $$400,$ and you expect it to last for $10$ years. The interest rate is $5\%.$

$($a$)$ Provide the formula that calculate the present value of the electricity bill savings you will receive over the next $10$ years $($Be careful for designating the final year$)$

$($b$)$ Provide the formula that calculate the net present value of the purchase