Trust Fund problem using series and sequences

[Aristarchus_]

Homework Statement

A person sets up a fund of 3,000,000 dollars which is to pay out a certain amount every year in the future, the first time in three years. The fund has a yield of 4.5% per year. How big can the annual payments be?

Relevant Equations

Convergance of a geometric series: $S_n = \frac {a_1}{1-k}$, where k is the quotient , a_1 is the first term.
Sum of a geometric series: $S_n = a_1 \cdot \frac{k^{n}-1}{k-1}$
The correct answer given in the textbook is 147,423 dollars

I have tried inserting 0.955 in the above formula for the sum of a geometric series and setting it equal to 3,000,000 (S_n) with n =3. This did not work out well
My second attempt was, considering that the payment is paid every year in the future, to use the convergence formula. There k = 0.955 and S_n = 3,000,000, this gives me the wrong answer of 135,000 dollars annual payment...
What does "first time in three years" imply? And how is it connected to the geometric series formula?

 

[pasmith]

If the value of the fund at the start of year n is P_n, then what is its value P_{n+1} at the start of the next year, given that we have withdrawn a paymanet of A during the year and received interest at a rate r = 4.5\,\% on the balance?

The question doesn't give a set term for the annuity, so I guess it is looking for the maximum payment we can take without diminishing the value of the fund, which is the value of A which makes P_{n+1} = P_n. Since we don't start withdrawing payments until after year 3, this tells us the value of P_n to use to determine A.

 

[Aristarchus_]

If the value of the fund at the start of year n is P_n, then what is its value P_{n+1} at the start of the next year, given that we have withdrawn a paymanet of A during the year and received interest at a rate r = 4.5\,\% on the balance?

The question doesn't give a set term for the annuity, so I guess it is looking for the maximum payment we can take without diminishing the value of the fund, which is the value of A which makes P_{n+1} = P_n. Since we don't start withdrawing payments until after year 3, this tells us the value of P_n to use to determine A.

What does "first time in three years" imply? I saw a couple of problems of this sort and it confuses me

 

[Mark44]

What does "first time in three years" imply?

Not sure what you mean by what it implies, but what it means is that the initial amount, $3,000,000, sits in the account for three years, accruing interest. No payments are withdrawn during this period.

 

[Aristarchus_]

Not sure what you mean by what it implies, but what it means is that the initial amount, $3,000,000, sits in the account for three years, accruing interest. No payments are withdrawn during this period.

Right...so it would grow by an annual interest amount?

 

[Aristarchus_]

Not sure what you mean by what it implies, but what it means is that the initial amount, $3,000,000, sits in the account for three years, accruing interest. No payments are withdrawn during this period.

I got the correct answer by setting in the geometric sum formula 3,000,000 * 1.045^2. This means, as you said, that in those three years, the amount has grown by this much (the first year just 3,000,000 fixed, second 3,000,000*1.045, third 3,000,000*1.045...). Thank you