How to calculate continuously compounded interest

Fixed deposit calculations can be done by both manual and automatic methods.Now let us figure out the fd calculation in both ways:
1.) Manual Calculation Technique: It is the conventional process to know your Fixed deposit interest calculation. The formula for manual fd calculation is used as below:

Where,A = Final Amount
P = Principal Amount (initial investment)
r = Annual Nominal Interest Rate (as a decimal, not in percentage)
n = Number of times the interest is compounded per year
t = Number of years

2.) Online Calculator Technique: Nowadays online FD calculator is very amazing technique and it is a very useful online tool that helps you to easily calculate your Fixed deposit return. Online FD calculators uses simple technique in which you need to put your amount, tenure, interest rate and can get accurate results instantly. You can easily calculate and compare the interest receivable by changing the deposit amount.

Well if you started with zero in the first year, then $164494, of which $30000 was yours, so earnings are 134494.

If you started with $1000, then $181943, of which $31000 was yours, so earnings are $150943

Since you didn't specify what result you were expecting, I can only guess at what you want.

The answer is correct assuming 12 periods per year. To have 500 increase to 10000 in 30 months, you need an annual interest rate of 126%. For a more realistic result, set the number of periods per year to one by pressing 1 SHIFT [P/YR] then repeat the calculation. This will produce 10.5% for the annual interest rate.

Alternatively, if the interest is compounded monthly, specify 360 for N. This will give a value closer to 10% annually, compounded monthly.

Your result is for the 6.75% interest compounded monthly. The problem states that the interest is compounded semiannually. This makes a difference in the effective interest rate.

A 6.75% APR compounded semiannually gives an effective interest rate of about 6.864%:
Press 2 , 6 . 7 5 2nd >EFF

Converting this to APR gives about 6.657%:
Press 1 2 , 6 . 8 6 4 2nd >APR

If you use 6.657 for the interest rate instead of 6.75 you should get the correct result.

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If the interest is compounded monthly:

2nd [CLR TVM] (clear any existing results)
1 5 0 0 0 _/- PV (present value, negative because you're paying it out)
6 I/Y (annual interest rate)
25 2nd [*P/Y] N (25 years)
CPT FV (compute future value, see 66,974.55)

If the interest is compounded annually:

2nd [CLR TVM] (clear any existing results)
1 5 0 0 0 _/- PV (present value, negative because you're paying it out)
6 I/Y (annual interest rate)
2nd [P/Y] 1 ENTER 2nd [QUIT] (one compounding period per year)
25 N (25 years)
CPT FV (compute future value, see 64,378.96)

Press APPS and select the Finance app. Select the TVM Solver.

Enter -4000 for PV, 0 for PMT, 6000 for FV. Be sure you use the (-) key for PV, not the - key. P/V and C/Y should both be 1 and PMT should be END.

If you want 5% annual interest compounded month, enter 5/12 for I%. If you simply want 5% annual interest, enter 5 for I%. Go to N and press ALPHA ENTER to see the number of periods (months or years according to what you entered earlier in this paragraph).

=10000*(1+0.96)^12
=10000*(1+0.10)^18
=10000*(1+0.10)^24