Final answer:
To solve the problem, use the compound interest formula to find the number of years. Then, use the same formula to find the amount and subtract the initial investments to find the total interest earned.
Step-by-step explanation:
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount
- P is the principal amount (the initial investment)
- r is the annual interest rate (as a decimal)
- n is the number of times that interest is compounded per year
- t is the number of years
In this case, Gary invested $1,000 and each year he deposited an additional $1,000. The balance after some years was $4,525.64. We need to find the value of t, the number of years. We can rearrange the formula to solve for t:
t = log(A/P) / log(1 + r/n)
Substituting the given values, we have:
t = log(4525.64/1000) / log(1 + 0.05/1)
Using a calculator, the value of t is approximately 4.999 years. This means the money was in the account for about 5 years.
To find out how much interest Gary would earn if he invests $10,000 and deposits $10,000 into the account each year for the same amount of time, we can use the same formula. Substituting the new values, we have:
A = 10000(1 + 0.05/1)^(1*5) + 10000(1 + 0.05/1)^(2*5) + ... + 10000(1 + 0.05/1)^(5*5)
Simplifying this expression using a calculator will give you the total amount, and subtracting the initial investments will give you the total interest earned.