Is (11,13,17) the only set of primes where the sum of the last 2 divided by the first is a recurring decimal?

Let's test with the next set:

(13, 17, 19)

(17 + 19) / 13

36/13

It's a rational number where 13 is not a factor of 36, so it will be a repeating decimal.

So your statement is false.

27 is not prime....

41+43=84

36/13=2.769230 recurring

As long as the first prime isn't 2 or 5 (because 2 and 5 are factors of our base-10/decimal numbering system) and isn't a factor of the sum of the second two primes, you'll have a recurring decimal. The only thing is the period of the repeat can be large and hard to see.

Example:

7, 11, 13

24/7 = 3.428571 428571...

23, 27, 29

56/23 = 2.(22 digits) (22 digits)...

37, 41, 43

88/37 = 2.270 270...

101, 103, 107

210/101 =  2.0792 0792...