Letter pattern with alternating swaps: complete the sequence FAG, GAF, HAI, IAH, ____ by recognizing swap/no-swap alternation and +1 increments across first and third letters

Given data

• Series: FAG, GAF, HAI, IAH, ____.

Concept/Approach (why this method)

Observe two simultaneous rules: (i) the first and third letters each increase by +1 per step overall (net +2 every two terms), and (ii) terms alternate between 'normal order' and 'swapped' order of the first/third letters.

Step-by-Step calculation
1) T1 = F A G (normal order).2) T2 = G A F (swap of T1's 1st and 3rd).3) T3 = H A I (both letters +2 from T1: F→H, G→I; normal order).4) T4 = I A H (swap of T3).5) T5 must be normal order again with +2 from T3: H→J and I→K ⇒ J A K.

Verification/Alternative

Check increments: (F,G)→(H,I)→(J,K) in normal terms; and each alternate term swaps those two letters, matching observed T2 and T4.

Common pitfalls

• Applying +1 rather than the two-step +2 across non-adjacent normal terms.
• Forgetting the swap/no-swap alternation.

Final Answer
JAK