Lottery.$$Your dreams of becoming rich have just come true. You have won the State of$$ Tranquility's Lottery. The State offers you two payment plans for the $$$$2$ comma $000$ comma $000$ advertised jackpot. You can take annual payments of $$$$100$ comma $000$ at the end of the year for the next $20$ years or $$$$796$ comma $333$ today.

a$.$If your investment rate over the next $20$ years is $12\%,$ which payoff will you$$ choose?

b$.$ If your investment rate over the next $20$ years is $9\%,$ which payoff will you$$ choose?

c$.$ At what investment rate will the annuity stream of $$$$100$ comma $000$ be the same as the$$ lump$-$sum payment of $$$$796$ comma $333?$

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Part $1$

a$.$If your investment rate over the next $20$ years is $12\%,$ what is the present value of the $$$$100$ comma $000$ annual payments$$ today?b