A level maths question help

I'm struggling to answer this A level maths question.
A savings scheme is offering a rate of interest of 3.5% per annum for the lifetime of the plan. Alan wants to save up £20000. He works out that he can afford to save £500 every year, which he will deposit on 1st January. If interest is paid on 31st December, how many years will it be before he has saved up his £20000?
I looked at the solution bank on Physics and maths tutor and was quite confused by the solution (see images). I'm sure why a is 1.035 I can see why r is 1.035 but not why a would also be 1.035. I then don't understand why you multiply by 500 either and am overall just confused. Could someone please help and explain to me what's happening and why so i understand it and can apply it to future questions. Thanks!

Its compound interest with deposit, so pmt writes out what happens in year 1, 2, ... and then writes it as a general expression in terms of year n. Do you understand the year 1, 2, ... stuff? If not, what do you think happens?

I understand the initial thing about writing about the year 1 and 2 stuff for the first 2 years i thought of that kind of equation being ((500x1.035)+500)x1.035 but after that im completely lost. I am familiar with geometric series and can use the formula confidently for other questions but this one is confusing me

If you cant see the general formula from the years 1 and 2, what do you get for year 3, year 4 ...

i can only really think of adding more brackets for example doing this for year 3 (((500x1.035)+500)x1.035)+500)x1.035

and when you expand that you get ....
You should be able to spot / understand the general formula for compound interest with deposits. So at year n, you have the sum of
this years deposit: 500*1.035
last years deposit: 500*1.035^2
2 years ago deposit: 500*1.035^3
....

ahh yes okay i see its kinda like normal compound interest say for example after 20 years the origonal 500 would be 500x1.035^20 . What happens with adding the £500 each year tho?

See the previous post? If n was 20, the deposit 20 years ago would now be worth
500*1.035^20
the deposit 19 years ago would now be worth
500*1.035^19
....
the depost last year would now be worth
500*1.035^2
and this years deposit would be worth
500*1.035
So simply sum them up and replace 20 with n.

ahhh okay i seee that makes a lot of sense im understanding that part now and i understand the next line in the working how they put the 500 outside of the bracket. Im still confused about the bits after this but im understanding everything up until now.

The bit in the bracket is a geometric series with a=1.035, r=1.035 and then just use the sum to n then logs to solve for n.