A hockey arena has a total seating capacity of 15 960. The first
row of seats around the rink has 262 seats. The number of seats
in the subsequent row increases by 18.


a) How manny rows in the arena?


total = row1 + (row1 + inc*1) + (row1 + inc*2) + ... + (row1 + inc*(n-1))
total = row1 * n  +  (inc*1) + (inc*2) + ... + (inc*(n-1))
total = row1 * n  +  inc * [ 1 + 2 + ... + (n-1) ]

    We know that sum_up_to(i) = 1/2 * i * (i+1)

    so
    sum_up_to(n-1) = 1/2 * (n-1) * ((n-1)+1)
    sum_up_to(n-1) = 1/2 * (n-1) * n

total = row1 * n  +  inc * [ 1/2 * (n-1) * n ]
total = row1 * n  +  1/2 * inc * (n-1) * n
total = row1 * n  +  1/2 * inc * (n^2 + -n)
total = row1 * n  +  1/2 * inc * n^2  +  -1/2 * inc * n
total = row1 * n  +  1/2 * inc * n^2  +  -1/2 * inc * n
[ 1/2 * inc * n^2 ]  +  [ -1/2 * inc * n  +  row1 * n ]  +  [ -total ] = 0
[ 1/2 * inc ] * n^2  +  [ -1/2 * inc + row1 ] * n  +  [ -total ] = 0

    total = 15960
    row1  = 262
    inc   = 18

[ 1/2 * 18 ] * n^2  +  [ -1/2 * 18 + 262 ] * n  +  [ -15960] = 0
9*n^2 + 253*n + -15960 = 0

    Solve for n, work ommited...

n = 30.33917358325823
  and/or
n = -58.450284694369344


Obviously, we disregard the negative result, so we
need more than 30 rows. The answer is therefore 31.


[[[[ The number of rows is 31. ]]]]



b) If three more rows of seats were added after the last row
   maintaining the same pattern, by what % would the seationg
   capacity increase?


[ It's not clear if I should use 15960 or the real seating ]
[ limit of 31 seats. I'll use the real seatting limit.     ]


From above,
total = row1 * n  +  1/2 * inc * (n-1) * n

    row1  = 262
    inc   = 18
    n     = 31

old_total = 262 * 31  +  1/2 * 18 * (31-1) * 31
old_total = 16492

    row1  = 262
    inc   = 18
    n     = 31 + 3 = 34

new_total = 262 * 34  +  1/2 * 18 * (34-1) * 34
new_total = 19006


old_total * (1 + percent) = new_total
percent = (new_total / old_total) - 1
    
    old_total = 16492
    new_total = 19006

percent = (19006 / 16492) - 1
percent = 0.15244
percent = 15%

[[[[ It's an increase of 15% ]]]]