Suppose that P dollars is invested in a savings account at interest rate I, compounded semiannually, for one year. The amount A in the account after one year is given by A = P(1 + i/2)^2
To find the amount A in the account after one year, we can use the formula A = P(1 + i/2)^2, where P is the principal amount (the initial investment), i is the interest rate, and A is the final amount in the account after one year.
Here's how you can calculate the final amount A:
1. Plug in the values of P (the principal amount) and i (the interest rate) into the formula.
2. Divide the interest rate i by 100 to convert it from a percentage to a decimal.
3. Add 1 to the interest rate decimal, then divide it by 2. This represents the interest rate for each compounding period (semiannual in this case).
4. Raise the result from step 3 to the power of 2, since the formula is compounded semiannually for one year.
5. Multiply the result from step 4 by the principal amount P to calculate the final amount A.
Let's go through an example: suppose $1000 is invested in a savings account with an interest rate of 5% compounded semiannually for one year.
1. P = $1000 and i = 5%
2. i/100 = 0.05
3. (1 + 0.05/2) = 1.025
4. (1.025)^2 = 1.050625
5. A = 1.050625 * $1000 = $1050.63
So, after one year, the amount in the account would be $1050.63.