Final answer:
- a) The amount in the account immediately after Juan's 35th deposit is approximately $10,629.47.
- b) Juan would need to invest approximately $4,723.59 on his 36th birthday to have the same account balance on his 60th birthday.
- c) A uniform annual investment of approximately $381.28 is required to achieve the same account balance.
Step-by-step explanation:
**a) In this investment scenario, Juan deposits $5,500 on his 26th birthday, and each year thereafter, he deposits 9% more than the previous deposit. The account pays annual compound interest of 3%. To calculate the amount in the account immediately after his 35th deposit, we need to determine the total amount of each deposit and the interest earned. The formula for compound interest is:
A = P(1 + r/n)⁽ⁿᵗ⁾
where A is the final amount, P is the principal (initial deposit), r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.
In this case, the principal is $5,500, the interest rate is 3%, compounded annually, and the number of years is 35. Using the formula, we can calculate the final amount after 35 years:
A = 5500(1 + 0.03/1)⁽¹*³⁵⁾
A = $5500(1.03)³⁵
A ≈ $10,629.47
Therefore, the amount in the account immediately after Juan's 35th deposit is approximately $10,629.47.
**b) If Juan decides to wait 10 years before investing for retirement, he would need to invest a different amount on his 36th birthday to have the same account balance on his 60th birthday. To calculate this amount, we can work backward from the desired account balance on his 60th birthday.
Using the same formula as before, A = P(1 + r/n)⁽ⁿᵗ⁾, we can substitute the values for the desired account balance, the interest rate, the compounding period, and the number of years to solve for P (the principal) on the 36th deposit:
$10,629.47 = P(1 + 0.03/1)⁽¹*²⁴⁾
Simplifying the equation:
10,629.47 = P(1.03)²⁴
P = 10,629.47 / (1.03)²⁴
P ≈ $4,723.59
Therefore, Juan would need to invest approximately $4,723.59 on his 36th birthday to have the same account balance on his 60th birthday.
**c) To determine the uniform annual investment required to achieve the same account balance, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)ⁿ - 1) / r
where FV is the future value, P is the periodic payment (uniform annual investment), r is the interest rate, and n is the number of compounding periods.
In this case, we want to solve for P:
$10,629.47 = P * ((1 + 0.03)³⁵ - 1) / 0.03
Simplifying the equation:
10,629.47 = P * (1.03³⁵ - 1) / 0.03
P ≈ $381.28
Therefore, a uniform annual investment of approximately $381.28 is required to achieve the same account balance.
Your question is incomplete, but most probably the full question was:
On Juan's 26th birthday, he invested $5,500 in a retirement account. Each year thereafter, he deposited 9% more than the previous deposit. The account paid annual compound interest of 3%.
- a) How much was in the account immediately after his 35th deposit?
- b) If Juan decided to wait 10 years before investing for retirement, how much would he have to invest on his 36th birthday to have the same account balance on his 60th birthday?
- c) What uniform annual investment is required to achieve the same account balance?