Calculating Monthly Investment

If I want to obtain a future value of $500,000 at the end of 20 years, how much do I need to save each year at an interest rate of 10% per annum? I ended up using the following formula: =PMT(10%,20,-PV(10%,20,,-500000,1),,1). Let's define (Insert/ Name/Define) the answer to this function as Pmt1.

By then using =FV(10%,20,Pmt1,,1) to confirm that Pmt1 will end up providing $500,000 after 20 years I get the answer I am looking for but have absolutely NO IDEA why it works. Worse, I do not know whether it is the correct answer. I have the following function (courtesy of someone) that I use to determine the expected future value of a series of annual payments at a fixed interest rate but also with fixed annual increases in the payments. (Example: $1000 per annum is invested for 20 years. The interest earned on the $1000 is 10% per annum. The $1000 increases by 5% each year - i.e. 19 increases)

=Pmt1* SUMPRODUCT((1+5%)^(ROW(OFFSET($A$1,0,0,20,1))-1),(1+10%)^(20-ROW(OFFSET($A$1,0,0,20,1))+1))


Assuming the payment does not increase, I simply replace the 5% with 0%. When I run this function and use Pmt1 as the annual payment the answer differs from the one that I get from the PMT function that I quoted above until I change the ",,1" in the function to ",,0". What do I not understand about these functions!? Which is correct or are both provided I learn to know what they do? This is the vaguest question I've ever been able to devise simply because I can see that something is amiss and I do not know what - or how to start figuring out what it is that I "know not"!