A 65 year matured man intends to use his retirement funds to purchase an annuity from a natural life insurance company.?

given the amount of money the man has available to invest, the insurance company is able to set aside two alternatives. the first option is to receive $2785 each month for as long as he lives; the second resort is to receieve $3500 each month, but for only 20 years (payments will be made to his estate if he should die past that time) the relevant interest rate is 6 percent per year. how long must the man live so that the first option is a better deal?

can someone plz relieve me answer this question for my finance assignment?
Answers:
Let:
p be the amount of respectively payment,
r be the fractional interest rate per year,
n be the number of years,
k be the number of payments and compounding periods per year,
s be the sum invested.

The present attraction of payments is the sum of a geometric series with first term p and adjectives ratio (1 + r / k)^(-1):
s = p sum(i = 0 to nk - 1) (1 + r / k)^(-ki)
s= p(1 - (1 + r / k)^(-nk)) / (r / k)
s = (pk / r)(1 - (1 + r / k)^(-nk)) ...(1)

Solving for n:
sr / kp = 1 - (1 + r / k)^(-nk)
1 - sr / kp = (1 + r / k)^(-nk)
log(1 - sr / kp) = - nk log(1 + r / k)
n = - log(1 - sr / kp) / k log(1 + r / k) ...(2)

For the 20 year option, (1) gives:
s = (3500*200)(1 - 1.005^(-240))
s = $488,532.70

For the lifetime choice, (2) gives:
n = - log(1 - 488,532.70 * 0.06 / (12 * 2785)) / 12 log(1.005)
n = 35.02yr.

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