Solution below.

Reveal solution by highlighting:

Adding the first 6 even numbers: 2+4+…+12 = 2*(1+2+…+6) = 2*(6*(1+6)/2) = 6*(1+6). Then we add half the next even number, 14/2 = 7. This turns 6*7 into 7*7 = 49.

Adding the first n even numbers: 2+4+…+2n = 2*(1+2+…+n) = 2*(n*(1+n)/2) = n*(1+n). Then we add half the next even number, (2n+2)/2 = n+1. This turns n*(1+n) into (1+n)*(1+n) = (1+n)².

(1+n)² = 465124 = 682².

1+n = 682 <=> n = 681.

(Exclude the solution where n < 0.)