Decoding the Letter-Number Series Logic

The question presents a series of letter-number clusters: DEW30, JKC45, PQI60, VWO75, ?. We need to find the pattern governing this series to determine the missing cluster.

Let's break down the series into its components: the first letter, the second letter, the third letter, and the number.

Analyzing the First Letter Pattern

Consider the first letter of each cluster: D, J, P, V.

• The position of D in the alphabet is 4.
• The position of J in the alphabet is 10.
• The position of P in the alphabet is 16.
• The position of V in the alphabet is 22.

Let's find the difference between these positions:

• $10 - 4 = 6$
• $16 - 10 = 6$
• $22 - 16 = 6$

The pattern is an addition of 6 to the alphabet position. To find the next letter's position:

• $22 + 6 = 28$

Since there are only 26 letters in the alphabet, we wrap around. The 28th position corresponds to the 2nd letter ($28 - 26 = 2$). The 2nd letter is B.

Analyzing the Second Letter Pattern

Now let's look at the second letter of each cluster: E, K, Q, W.

• The position of E is 5.
• The position of K is 11.
• The position of Q is 17.
• The position of W is 23.

Calculate the differences:

• $11 - 5 = 6$
• $17 - 11 = 6$
• $23 - 17 = 6$

This pattern also involves adding 6 to the previous letter's position. For the next letter:

• $23 + 6 = 29$

Wrapping around the alphabet, the 29th position corresponds to the 3rd letter ($29 - 26 = 3$). The 3rd letter is C.

Analyzing the Third Letter Pattern

Let's examine the third letter of each cluster: W, C, I, O.

• The position of W is 23.
• The position of C is 3.
• The position of I is 9.
• The position of O is 15.

Calculate the differences, remembering to wrap around the alphabet:

• From W (23) to C (3): $23 \rightarrow 24 \rightarrow 25 \rightarrow 26 \rightarrow 1 \rightarrow 2 \rightarrow 3$. This is a step of 6.
• From C (3) to I (9): $9 - 3 = 6$.
• From I (9) to O (15): $15 - 9 = 6$.

The pattern is consistently adding 6. For the next letter:

• $15 + 6 = 21$

The 21st letter of the alphabet is U.

Analyzing the Number Pattern

Finally, let's analyze the number part of the clusters: 30, 45, 60, 75.

Calculate the difference between consecutive numbers:

• $45 - 30 = 15$
• $60 - 45 = 15$
• $75 - 60 = 15$

The pattern is adding 15 to the previous number. To find the next number:

• $75 + 15 = 90$

Determining the Missing Cluster

Combining the patterns we found:

• First Letter: B
• Second Letter: C
• Third Letter: U
• Number: 90

Therefore, the missing cluster that logically completes the series is BCU90.