When Alfred Nobel$$ died, he left the majority of his estate to fund five$$ prizes, each to be awarded annually in perpetuity starting one year after he died.$($The sixth$$ one, in$$ economics, was added$$ later.$)$

a$.$ If he wanted the cash award of each of the five prizes to be $$$$55$ comma $000$ and his estate could earn $9\%$ per$$ year, how much would he need to fund his$$ prizes?

b$.$ If he wanted the value of each prize to grow by $4\%$ per year$($perhaps to keep up with$$ inflation$),$ how much would he need to$$ leave? Assume that the first amount was still $$$$55$ comma $000.$

c$.$ His heirs were surprised by his will and fought it$.$ If they had been able to keep the amount of money you calculated in b and had invested it at $9\%$ per$$ year, how much would they have in $2013,117$ years after he$$ died?