8 11 21 15 18 21 22

Concept:

A number series that mixes a repeated main-sequence rule with an inserted constant term has two separate layers: an underlying main sequence built by a repeating arithmetic operation (here, addition that alternates between two fixed amounts), and a fixed extra number reinserted at a regular interval, which does not itself take part in that arithmetic rule. Solving such a series means separating the two layers, finding the rule for the main sequence, and confirming exactly when the inserted constant recurs.

Applying it here:

1. Separate the interpolated terms: removing the repeated 21s from 8, 11, 21, 15, 18, 21, 22 leaves the main sequence 8, 11, 15, 18, 22.

2. Find the rule governing the main sequence: 11 − 8 = 3, 15 − 11 = 4, 18 − 15 = 3, 22 − 18 = 4 — the addition alternates between +3 and +4.

3. Locate where the constant 21 is inserted: it appears right after 11 (reached by +3) and right after 18 (reached by +3), but not after 15 or 22 (each reached by +4) — so 21 is inserted specifically after a term reached via a +3 step.

4. Continue the main sequence from 22: since the previous step was +4, the alternation calls for +3 next, giving 22 + 3 = 25.

5. Since 25 was reached via a +3 step, the inserted constant follows immediately after it, giving 21 as the next term.

Cross-check:

The insertion rule holds consistently across every occurrence in the given data — 21 follows only the terms reached by +3 (11, 18), never the terms reached by +4 (15, 22) — so applying the same rule to the new +3 result (25) is consistent with the entire pattern already shown, not an isolated guess.

The next two terms of the series are therefore 25 followed by 21.