Assume that today is January $1,$ year $1.$ Given a discount rate of $8\%,$ calculate the following values.

i$.$ Present value of a perpetuity $($also called a perpetual annuity$)$ of $$52$ received each year at the end of each year $($i$.$e$.,$ each December $31$ from now to the end of time$)$

ii$.$ Divide the perpetuity from part $($i$)$ into three parts, and calculate the present value today of each part:

Part $1$: an annuity of $$52$ received at the end of each year for five years $($i$.$e$.,$ each December $31$ from year $1$ to year $5)$

Part $2$: an annuity of $$52$ received at the end of each year for $10$ years $($i$.$e$.,$ each December $31$ from year $6$ to year $15)$

Part $3$: a perpetuity of $$52$ received at the end of each year from year $16$ to the end of time $($i$.$e$.,$ each December $31$ from year $16$ and forever thereafter$)$

iii. Add the parts of $($ii$)$ together, and ensure that the total is the same as the value calculated in part i$.$