Explanation:

- Let’s first find the pattern in the series: 58, 73, 99, 138, 193, ?, 368

- Find the difference between consecutive numbers:

- 73 - 58 = 15

- 99 - 73 = 26

- 138 - 99 = 39

- 193 - 138 = 55

- The differences are: 15, 26, 39, 55

- Now check the pattern in the differences:

- 26 - 15 = 11

- 39 - 26 = 13

- 55 - 39 = 16

- The second-level differences: 11, 13, 16 (increasing by 2, 3…)

- Next difference (after 55) should be: 55 + (16 + 3) = 55 + 19 = 74

- Add 74 to previous term: 193 + 74 = 267

- Next: 368 - 267 = 101 (for confirmation: 74 + difference, 19 + (3) = 22; 74+22=96. Hmm, progression might be +2, +3, +3, try with +3 next.)

Let’s check again in steps for correctness:

- 15, (add 11=) 26, (add 13=) 39, (add 16=) 55.

- Now, 16 was added previously, likely to continue with +3, so next should be 16 + 3 = 19; 55 + 19 = 74.

- 193 + 74 = 267.

- 368 - 267 = 101.

- Let's check options:

- Option 1: 260 (193+67 = 260, difference not matching pattern)

- Option 2: 268 (193 + 75 = 268, but previous pattern was +74)

- Option 3: 312 (193+119=312, way off)

- Option 4: 329 (193+136=329, mismatch)

Conclusion:

- The correct next value is 267 (not in the options).

- Closest option is Option 2: 268.

Correct Option: None perfectly matches, but Option 2 (268) is closest to the correct answer 267.

- Series uses an increment with growing step: +15, +26, +39, +55, +74.

- Next should be 267 (193+74), but as options skip this, Option 2 is nearest.

- Option 2 (268) is close, but not exact.

- The exact answer should have been 267 per the established pattern.