A bank account was opened with a deposit of $2750 at an APR of 3.5% compounded one of the following ways. Find the amount in this account after 6 years with compounding of:
(round answers to nearest 100th)
annually: $
semi-annually: $
quarterly: $
monthly: $
weekly: $
daily: $
1000 times a year: $
10,000 times a year: $
r = .035/n where n = times per year
yearly gain = 1+r
final amount = 2750 * (1+r)^6n
so for example for 1000 times per year:
r = .035/1000 = .000035
1+r = 1.000035
and amount = 2750 *(1.000035)^6000
= $ 3392.60
gain per period = 1+r
not per year
gain per year = (1+r)^n
gain per 6 years = (1+r)^6n
To find the amount in the account after 6 years with different compounding frequencies, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the initial deposit
r = the annual interest rate (expressed as a decimal)
n = the number of compounding periods per year
t = the number of years
Let's calculate the amount in the account after 6 years for each compounding frequency:
Annually (n = 1):
A = 2750(1 + 0.035/1)^(1*6)
A ≈ 2750(1.035)^6
A ≈ $3,147.36
Semi-annually (n = 2):
A = 2750(1 + 0.035/2)^(2*6)
A ≈ 2750(1.0175)^12
A ≈ $3,156.92
Quarterly (n = 4):
A = 2750(1 + 0.035/4)^(4*6)
A ≈ 2750(1.00875)^24
A ≈ $3,161.47
Monthly (n = 12):
A = 2750(1 + 0.035/12)^(12*6)
A ≈ 2750(1.002917)^72
A ≈ $3,163.62
Weekly (n = 52):
A = 2750(1 + 0.035/52)^(52*6)
A ≈ 2750(1.000673)^312
A ≈ $3,164.29
Daily (n = 365):
A = 2750(1 + 0.035/365)^(365*6)
A ≈ 2750(1.000095)^2190
A ≈ $3,164.57
1000 times a year (n = 1000):
A = 2750(1 + 0.035/1000)^(1000*6)
A ≈ 2750(1.000035)^6000
A ≈ $3,164.86
10,000 times a year (n = 10000):
A = 2750(1 + 0.035/10000)^(10000*6)
A ≈ 2750(1.0000035)^60000
A ≈ $3,164.93
Therefore, the amount in the account after 6 years, rounded to the nearest hundredth, is as follows:
Annually: $3,147.36
Semi-annually: $3,156.92
Quarterly: $3,161.47
Monthly: $3,163.62
Weekly: $3,164.29
Daily: $3,164.57
1000 times a year: $3,164.86
10,000 times a year: $3,164.93