finite math

A $1.2 million state lottery pays $5,000 at the beginning of each month for 20 years. How much money must the state actually have in hand to set up the payments for this prize if money is worth 7.7%, compounded monthly?
(a) Decide whether the problem relates to an ordinary annuity or an annuity due.

1
annuity due
ordinary annuity
.
(b) Solve the problem.

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  1. 1.
    Note: "...pays $5,000 at the beginning of each month..."
    Should that ring a bell?

    2.
    To find the present value of the annuity due, we can consider it as the sum of the first payment (immediate) and an ordinary annuity with one payment less (the last one).

    Thus:
    n=20*12-1=239
    R=5000
    i=7.7%/12

    P=R + R(1-(1+0.077/12)^(-239))/(0.077/12)
    =5000+5000(1-0.21682)/0.006416667
    =5000+5000*122.0539
    =5000+610269.5
    =$615,269.50

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