Brenda invests $1,000 every 6 months in a savings account with a fixed annual interest rate of 5% compounded 2 times per year. What will the account balance be after 6 years?
To calculate the account balance after 6 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the account balance after a certain period of time
P = the principal amount (initial investment)
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, Brenda invests $1,000 every 6 months, so we need to calculate the account balance for each 6-month period and sum them up.
First, let's calculate the account balance for the first 6-month period:
Principal (P) = $1,000
Annual interest rate (r) = 5% = 0.05 (converted to decimal form)
Number of times compounded per year (n) = 2 (compounded 2 times per year)
Time (t) = 0.5 years (6 months)
Using the compound interest formula:
A1 = $1,000(1 + 0.05/2)^(2*0.5)
= $1,000(1 + 0.025)^1
= $1,000(1.025)
= $1,025
Now, let's calculate the account balance for the second 6-month period:
Principal (P) = $1,000
Annual interest rate (r) = 5% = 0.05 (converted to decimal form)
Number of times compounded per year (n) = 2 (compounded 2 times per year)
Time (t) = 0.5 years (6 months)
Using the compound interest formula:
A2 = $1,025(1 + 0.05/2)^(2*0.5)
= $1,025(1 + 0.025)^1
= $1,025(1.025)
= $1,050.62
Now, let's calculate the account balance for the third 6-month period:
Principal (P) = $1,000
Annual interest rate (r) = 5% = 0.05 (converted to decimal form)
Number of times compounded per year (n) = 2 (compounded 2 times per year)
Time (t) = 0.5 years (6 months)
Using the compound interest formula:
A3 = $1,050.62(1 + 0.05/2)^(2*0.5)
= $1,050.62(1 + 0.025)^1
= $1,050.62(1.025)
= $1,076.89
We repeat this process for each 6-month period for a total of 12 times (6 years multiplied by 2 times compounded per year).
After calculating the account balance for 12 periods, we can sum up the results to find the final account balance after 6 years:
Final Account Balance = A1 + A2 + A3 + ... + A12
Performing the calculations and summing up the results will give us the answer.