Claire has borrowed $\$5,\!000$. She plans to pay off the loan in full after two payments. She will make one payment in 3 years, then another payment in 6 years. The second payment will be exactly double the amount of the first payment. How much is the first payment if the interest rate of the loan is $8.5\%$, compounded annually? Express your answer as a dollar value rounded to the nearest cent.
Don't try fancy codes here, they don't work.
I will read that as $5,000
Pick your "time spot" at 6 years, then
2x + x(1.085)^3 = 5000(1.085)^6
x(2 + 1.08r^2) = 8157.34
x = 8157.34/3.277289... = $2489.05
lmfao no ur not and aops administrator stop the cap
Let Claire's first payment be $x$, so her second one is $2x$. The present value of the first payment is $(x)/(1.085)^3$ and the present value of the second is $(2x)/(1.085)^6$. The sum of these present values must equal $\$5,\!000$, the amount Claire borrows today:\[\frac{x}{1.085^3} + \frac{2x}{1.085^6} = \$5,\!000 \implies x = \boxed{\$2,\!489.05}.\]
confirmed the first answer
did someone just pretend to say they're the aops admin this is serious cheating but i dont think they're the real aops admin but who knows
If it was an aops admin you could use a vpn
To find the amount of the first payment, we need to first calculate the total amount owed at the end of the 3-year period, including interest. We can then divide this amount by a factor to find the first payment.
1. Calculate the total amount owed at the end of the 3-year period:
Since the loan is compounded annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the total amount owed at the end of the period
P = the principal amount (initial loan amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, P = $5,000, r = 0.085 (8.5% expressed as a decimal), n = 1 (compounded annually), and t = 3.
Using these values, we can calculate the total amount owed at the end of the 3-year period:
A = 5000(1 + 0.085/1)^(1*3)
= 5000(1 + 0.085)^3
≈ 5000(1.085)^3
2. Calculate the second payment amount:
The second payment is exactly double the amount of the first payment, so:
Second payment = 2 * First payment
3. Calculate the first payment amount:
To find the first payment, we divide the total amount owed at the end of the 3-year period by a factor:
First payment = A / (1 + 1/(1 + r)^6)
In this case, A ≈ 5000(1.085)^3 and r = 0.085 (same as before).
Using these values, we can calculate the first payment:
First payment ≈ (5000(1.085)^3) / (1 + 1/(1 + 0.085)^6)
Calculating this expression, we find the first payment to be approximately $2324.41, rounded to the nearest cent.
Therefore, the first payment is approximately $\$2324.41.