Complete the given series by finding the missing term 28, 33, 31, ?, 34, 39

Concept: A number series that has no single constant difference may instead follow two different operations applied alternately in a fixed cycle (for example, add a fixed amount, then subtract a fixed amount, repeating). To find a missing term, first determine the two operations and their order from the terms already known, then continue applying the same cycle in sequence.

Application: Compute the difference between each pair of consecutive known terms to reveal the operating cycle.

1. 28 to 33: 33 − 28 = 5, so the first step in the cycle is +5.

2. 33 to 31: 31 − 33 = −2, so the second step in the cycle is −2.

3. The cycle +5, −2 repeats; the next step after the third term (31) is +5, so the missing term = 31 + 5 = 36.

Cross-check: Continuing the same cycle from the missing term confirms it fits: 36 − 2 = 34, then 34 + 5 = 39 — exactly matching the two terms given after the gap. This confirms the missing term is 36.