Ya...I see...a "constant principal" loan; yours will look like:

Code:

YEAR PAYMENT INTEREST BALANCE
0 500,000.00
1 -50,000.00 25,000.00 475,000.00
2 -48,750.00 23,750.00 450,000.00
3 -47,500.00 22,500.00 425,000.00
4 -46,250.00 21,250.00 400,000.00
5 -45,000.00 20,000.00 375,000.00
...
19 -27,500.00 2,500.00 25,000.00
20 -26,250.00 1,250.00 .00

The INTEREST column will add to 262,500;

Per your formula: 500000(.05)(21) / 2 = 262,500 ; so formula CORRECT!

The "principal" portion of the payment is simply A / n : 500000 / 20 = 25,000;

what is owing after x payments: A - x(A / n) ;

after 5 years (x = 5) : 500000 - 5(500000 / 20) = 375,000

A = 500000, P = 25000, i = .05, n = 20, x = 5

To calculate the interest paid after x payments:

[2A - P(x-1)] * i * x / 2

= 900000 * .05 * 5 / 2

= 112,500 (agrees with my illustration above; add the interest shown)

SOoooooo:

annuity basis 1st payment: ~40,121

serial basis 1st payment: 50,000

That's why total interest is lower with serial...or "appears" lower...