Finance problems (TVM) on hp 49G+ hp 50G Periods and compounds per year are different

[Gil]

For the same example
N=60
I%nominal=5%
PV=30000
FV=0
C/YR=3
P/YR=12

Suppose that you don't have the options C/YR and P/YR
on your calculator.

Then, instead of 5%, put the corresponding value x% calculated as follows:
0.05/3
0.05/3 +1
(0.05/3+1)^3
((0.05/3+1)^3)^(1/12)=1.00414087532
1.00414087532-1
x%=(1.00414087532-1)*100=0.414087532%
And you should get PMT=-565.711726.

Suppose, now, that you don't have any available program,
then procede as follows to calculate PMT:
PV*(... )+PMT*[... ]=FV=0

PV*(1.00414087532^60)
+PMT*[(1.00414087532^60-1)/0.00414087532)=0

30000*1.2813824439
+PMT*[(1.2813824439-1)/0.00414087532)=0

38441.473317
+PMT*[67.9524067] =0

PMT(at the end) =-38441.473317/[67.9524067] =565.711726
PMT (at the beginning) =PMT(at the end) / 1.00414087532
PMT (at the beginning) =-565.711726 /1.00414087532
PMT (at the beginning) = -563.378845

Balance after one full year = 12 months
(payments at the end of each month)

PV*(...)
+PMT*[...]

PV*(1.00414087532^12)
+PMT*[(1.00414087532^12-1)/0.00414087532]

PV*(1.05083796299)
+PMT*[0.05083796299/0.00414087532]

PV*(1.05083796299)
+PMT*[12.277105]

30000*(1.05083796299)
+-565.711726*[12.277105]

=24579.836

Within the first year, the debt went down from 30000 to 24579.836.
—> "Reduction" of 5420.164.
But we paid 12*PMT=12*-565.711726=6788.541.
—> Interests paid during the first year (12 full months)
=6788.541 - 5420.164 = 1368.377

Balance after exactly 12 payments
(but payments at the beginning of each month,
the 12th payment being made after exactly 11 months
[and not here after a full year=12 months]
—> balance "a few seconds" after the 11th month)

PV*(...)
+PMT*[...]/1.00414087532

PV*(1.00414087532^ 11 )
+PMT*[(1.00414087532^12-1)/(0.00414087532/1.00414087532)] /1.00414087532

PV*(1.00414087532^ 11 )
+PMT*[(1.00414087532^12-1)/0.00414087532]


PV*(1.0465045)
+PMT*[0.05083796299/0.00414087532]

PV*(1.0465045)
+PMT*[12.277105]

30000*(1.0465045)
+-563.3788*[12.277105]

31'395.1355-6916.6615
=24478.474