Arithmetic Progression (re-arranging equations)

[Mild illness]

I'm having problems with arithmetic progression you may have gathered, I have tried but I feel like I made it needlessly complicated as I am pretty bad at rearranging equations. I'll show the question and my attempt.

455 cylinders are stacked in rows in such a way that the in the row below the number is one less than the row above. The bottom row has 30 cylinders. How many rows are there and how many cylinders are in the top row.

I know the amount of rows is around 13 just through trial and error

I put everything I have to begin with so:
a = 30
d = 1
S_n=455

455 = ⁿ/₂[2a + (n-1)d]
910 = n[2a + (n-1)d]
-2n = 910/[2a -1x1] = 15.6 re-occurring (which is wrong, so I start from the line above again)

910 = n(61-n)
910 = 61n -n²
n² -61n + 910

[-b +/- sqroot(b² -4(ac)]/2a

[61 - sqroot(-61² - 4(1x910)]/2 = 26
26 is twice the answer I need and I don't think I can justify halving it.

The other attempts were me just rearranging the above equations in an attempt to get to the answer

 

[Mild illness]

Something wrong in problem: 455?
12 rows = 426
13 rows = 468

it should be 13 rows because it needs to say how many can accommodate all of them.
The second part of the question states how many cylinders are on the top row. It asks this because the row is not full up.
12 rows is too little if that makes sense. It should be 29 according to your numbers because 426 is not the full amount leaving 29 to be on the thirteenth row to make up the full 455, hope this helps
Sorry should have mentioned that.

 

[Mild illness]

I have worked it out

I shouldn't need any help I finally got to it after 3 hours and a painful headache