A benefactor leaves an inheritance to four charities, A, B, C, and D. The total inheritance is a series of level payments at the end of each year forever. During the first n years A, B, and C share each payment equally. All payments after n years revert to D. If the present values of the shares of A, B, C, and D are all equal, find (1 i) n .

(1+i)ⁿ = 4

Explanation:

there are 3 charities, and each charity receives X per year during n years. Total payment to the 3 charities is 3X per year.

The present value of the payments received by each charity is X · a_n (the same for A, B and C).

D will receive 3X at a deferred date starting at n. The present value of the payments that D receives is 3X · vⁿ · 1/i.

Since the present value of the payments is equal for all of them, then:

X · a_n = 3X · vⁿ · 1/i

a_n = 3/i · vⁿ

i · a_n = 3vⁿ

since i · a_n = 1 - vⁿ

1 - vⁿ = 3vⁿ

1 = 4vⁿ

(1 + i)ⁿ = 4

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