Gilberto opened a savings account for his daughter and deposited $1200 on the day she was born. Each year on her birthday, he deposited another $1200. If the account pays 9.5% interest, compounded annually, how much is in the account at the end of the day on her 10th birthday?
We can use the formula for compound interest:
A = P*(1+r/n)^(n*t)
where
A = the amount at the end of the investment period
P = the principal amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, P = $1200, r = 0.095, n = 1 (compounded annually), and t = 10.
For the first deposit, the amount after 1 year is:
A1 = 1200*(1+0.095/1)^(1*1) = $1314
For the second deposit, the amount after 2 years is:
A2 = (1200+1314)*(1+0.095/1)^(1*1) = $2871.30
For the third deposit, the amount after 3 years is:
A3 = (1200+1314+2871.30)*(1+0.095/1)^(1*1) = $4452.37
And so on, until we reach the 10th deposit and the amount after 10 years:
A10 = (1200+1314+2871.30+...+12207.63)*(1+0.095/1)^(1*10) = $29,992.60
Therefore, the amount in the account at the end of the day on her 10th birthday is $29,992.60.
To find out how much is in the account at the end of the day on her 10th birthday, we need to calculate the future value of the account.
The initial deposit of $1200 and the subsequent annual deposits of $1200 can be represented as an annuity. The annuity will compound interest at a rate of 9.5% annually.
The formula to calculate the future value of an annuity is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = periodic payment (annual deposit)
r = interest rate per period (9.5% = 0.095)
n = number of periods (10 years)
Plugging in the values into the formula, we get:
FV = 1200 * ((1 + 0.095)^10 - 1) / 0.095
Calculating this expression, we get:
FV ≈ $17,126.21
Therefore, the account will have approximately $17,126.21 at the end of the day on her 10th birthday.