# Trust Fund problem using series and sequences

• Aristarchus_
In summary, the question is asking for the answer to the following equation: P_n = A * (1 + r)n where r is the interest rate. Using the convergence formula, the answer is 135,000 dollars annual payment.

#### 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?

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at the start of year $n$ is $P_n$, then what is its value $P_{n+1}$ at the start of the next year, given that we have withdrawn a paymanet of $A$ during the year and received interest at a rate $r = 4.5\,\%$ on the balance?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

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at the start of year $n$ is $P_n$, then what is its value $P_{n+1}$ at the start of the next year, given that we have withdrawn a paymanet of $A$ during the year and received interest at a rate $r = 4.5\,\%$ on the balance?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

Aristarchus_ said:
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. Mark44 said: 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?

Mark44 said:
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