Understanding the Letter Cluster Series

The question asks us to identify the missing letter cluster in the given series: FKSN, HMQL, JOOJ, ?, NSKF. We need to find a logical pattern that connects these letter clusters.

Analyzing the Pattern in the Series

Let's break down the series and look for a pattern by comparing each letter's position in the alphabet.

We represent the letters by their corresponding numerical positions in the English alphabet:

• A=1, B=2, C=3, ..., Z=26

Let's convert each term in the series to its numerical equivalent:

• FKSN = 6, 11, 19, 14
• HMQL = 8, 13, 17, 12
• JOOJ = 10, 15, 15, 10
• ? = ?, ?, ?, ?
• NSKF = 14, 19, 11, 6

Step-by-Step Pattern Identification

Let's examine the relationship between consecutive terms:

Term 1 (FKSN) to Term 2 (HMQL)

• 1st letter: F (6) to H (8) is a change of $+2$ ($8 - 6 = 2$)
• 2nd letter: K (11) to M (13) is a change of $+2$ ($13 - 11 = 2$)
• 3rd letter: S (19) to Q (17) is a change of $-2$ ($17 - 19 = -2$)
• 4th letter: N (14) to L (12) is a change of $-2$ ($12 - 14 = -2$)

Term 2 (HMQL) to Term 3 (JOOJ)

• 1st letter: H (8) to J (10) is a change of $+2$ ($10 - 8 = 2$)
• 2nd letter: M (13) to O (15) is a change of $+2$ ($15 - 13 = 2$)
• 3rd letter: Q (17) to O (15) is a change of $-2$ ($15 - 17 = -2$)
• 4th letter: L (12) to J (10) is a change of $-2$ ($10 - 12 = -2$)

The observed pattern is consistent: Add 2 to the first two letter positions and subtract 2 from the last two letter positions to get the next term.

Calculating the Missing Term

We apply the identified pattern (+2, +2, -2, -2) to the third term (JOOJ) to find the fourth term:

Term 3: J (10), O (15), O (15), J (10)

• 1st letter: $10 + 2 = 12$. The 12th letter is L.
• 2nd letter: $15 + 2 = 17$. The 17th letter is Q.
• 3rd letter: $15 - 2 = 13$. The 13th letter is M.
• 4th letter: $10 - 2 = 8$. The 8th letter is H.

Therefore, the missing letter cluster is LQMH.

Verifying the Pattern with the Last Term

Let's check if applying the pattern to our calculated fourth term (LQMH) gives us the fifth term (NSKF).

Calculated Term 4: L (12), Q (17), M (13), H (8)

• 1st letter: $12 + 2 = 14$. The 14th letter is N.
• 2nd letter: $17 + 2 = 19$. The 19th letter is S.
• 3rd letter: $13 - 2 = 11$. The 11th letter is K.
• 4th letter: $8 - 2 = 6$. The 6th letter is F.

The result is NSKF, which matches the fifth term in the series. This confirms our calculation.

Conclusion

The pattern consistently adds 2 to the first two letters and subtracts 2 from the last two letters of each cluster to form the next cluster in the series. Applying this pattern to JOOJ yields LQMH.