Well, S(68) represents the sum of the arithmetic sequence from a(1) to a(68). To calculate that, we can use the formula for the sum of an arithmetic sequence: S(n) = (n/2)(a(1) + a(n)).
To find a(1), we substitute n = 1 into the given formula: a(1) = (2/3)(1) + 1/6 = 2/3 + 1/6 = 4/6 = 2/3.
Now, let's find a(68). Plugging n = 68 into the formula, we have a(68) = (2/3)(68) + 1/6 = 136/3 + 1/6 = 272/6 + 1/6 = 273/6.
Now, we're ready to calculate S(68). Plugging in a(1) and a(68) into the sum formula, we get:
S(68) = (68/2)(a(1) + a(68)) = (68/2)(2/3 + 273/6) = 34(2/3 + 273/6) = 34(8/6 + 273/6) = 34(281/6) = (34 * 281) / 6.
Doing the multiplication, we get: (34 * 281) / 6 = 9534 / 6 = 4,726/3.
So, S(68) is equal to 4,726/3.
Just like juggling numbers, that's quite a trick, huh?