If 3000 dollars is invested in a bank account at an interest rate of 6 per cent per year, find the amount in the bank after 12 years if interest is compounded annually

Find the amount in the bank after 12 years if interest is compounded quaterly
Find the amount in the bank after 12 years if interest is compounded monthly
Finally, find the amount in the bank after 12 years if interest is compounded continuously

12 years 3000 *1.06^12

12*4 = 48 quarters
.06/4 = .015
so
3000 * 1.015^48

12*12 = 144 months
.06/12 = .005
3000 * 1.005^144

dx/dt = .06 x
dx/x = .06 dt
ln x = .06 t
x = Xo e^.06 t
x = 3000 e^(.06*12)

To find the amounts in the bank after 12 years with different compounding periods, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years

Given:
P = $3000
r = 6% = 0.06
t = 12 years

1. Compounded annually (n = 1):
A = 3000(1 + 0.06/1)^(1*12)
A ≈ $5953.25

2. Compounded quarterly (n = 4):
A = 3000(1 + 0.06/4)^(4*12)
A ≈ $5961.68

3. Compounded monthly (n = 12):
A = 3000(1 + 0.06/12)^(12*12)
A ≈ $5966.76

4. Compounded continuously (n = infinity):
A = 3000e^(0.06*12)
A ≈ $5973.80

So, the amount in the bank after 12 years with different compounding periods would be approximately:
- Compounded annually: $5953.25
- Compounded quarterly: $5961.68
- Compounded monthly: $5966.76
- Compounded continuously: $5973.80

To find the amount in the bank after 12 years with different compounding periods, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = Final amount after interest
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years

1. Compounded Annually:
Using the given values:
P = $3000
r = 6% = 0.06 (converted to decimal)
n = 1 (compounded annually)
t = 12 years

Substituting these values into the formula, we have:
A = 3000(1 + 0.06/1)^(1*12)
A = 3000(1 + 0.06)^12
A ≈ $5743.49

Therefore, the amount in the bank after 12 years with annual compounding is approximately $5743.49.

2. Compounded Quarterly:
Using the same formula and values, with n = 4 (compounded quarterly):
A = 3000(1 + 0.06/4)^(4*12)
A = 3000(1 + 0.015)^48
A ≈ $5790.84

So, the amount in the bank after 12 years with quarterly compounding is approximately $5790.84.

3. Compounded Monthly:
Using the same formula and values, with n = 12 (compounded monthly):
A = 3000(1 + 0.06/12)^(12*12)
A = 3000(1 + 0.005)^144
A ≈ $5802.43

Thus, the amount in the bank after 12 years with monthly compounding is approximately $5802.43.

4. Compounded Continuously:
To calculate the amount with continuous compounding, we use the formula:

A = P * e^(rt)

Where:
e = Euler's number (approximately 2.71828)

Using the same values, we have:
A = 3000 * e^(0.06 * 12)
A ≈ $5814.10

Hence, the amount in the bank after 12 years with continuous compounding is approximately $5814.10.