When a list of numbers are equally spaced, their median equals their mean.

Their mean is their sum divided by the number of numbers.  There are 10
numbers, and their median (which is their mean) is 77, so their sum is 
77*10  or 770.

Or you can check it by going through the long process of writing them out:

Write 10 blanks for the consecutive even integers:

__, __, __, __, __, __, __, __, __, __

The median of an even number of number is the 
sum of the average of the middle-sized two.

The only two consecutive even integers that have average 77 are
76 and 78, because .
So we write 76 and 78 in the middle two blanks.

__, __, __, __, 76, 78, __, __, __, __
 
Then just fill in the rest by counting down from 76 by 2's to get the
ones below 76 and count up by 2's to get the ones above 78:

68, 70, 72, 74, 76, 78, 80, 82, 84, 86

Add them up and get 770.  But it's easier to reason it out immediately
as 770.

Edwin