Mary has a savings account and is earning 10% annual interest compounded semi-annually. If she deposits $2,000 in the account every six months for three years, what will the future value of her account be? How much interest will she have earned?
so, at 5% per period, after 6 periods, the account will have
2000*1.05^6 + 2000*1.05^5 + ... + 2000
= 2000(1.05^7 - 1)/0.05 = 16284.02
To find the future value of Mary's account and the interest earned, we can use the compound interest formula:
Future Value = Principal * (1 + Rate/100)^Time
where:
- Principal is the initial deposit
- Rate is the annual interest rate
- Time is the number of compounding periods
Let's break down the given information:
Principal: Mary deposits $2,000 in the account every six months, so she makes two deposits per year. This means her initial deposit is $2,000.
Rate: The annual interest rate is 10%, compounded semi-annually. Since it's compounded twice a year, we need to divide the annual rate by 2. Therefore, the rate per compounding period is 10% / 2 = 5%.
Time: Mary deposits money for three years, so she performs two deposits per year for a total of 3 years. Hence, the total number of compounding periods is 2 * 3 = 6.
Now, we can calculate the future value:
Future Value = $2,000 * (1 + 5%/100)^6
= $2,000 * (1.05)^6
≈ $2,000 * 1.340095
≈ $2,680.19
The future value of Mary's account after three years will be approximately $2,680.19.
To find the interest earned, we subtract the initial deposit from the future value:
Interest Earned = Future Value - Principal
= $2,680.19 - $2,000
≈ $680.19
Mary will have earned approximately $680.19 in interest.