Alphabet Series Pattern Analysis

The question asks us to find the next group of letters in the given alphabetical series: $PIK$, $IBD$, $BUW$, $UNP$, followed by a question mark ($?$).

To solve this, we need to identify the underlying pattern connecting these three-letter groups. We can do this by examining the relationship between the letters based on their positions in the English alphabet, where A = 1, B = 2, ..., Z = 26.

Decoding the Letter Sequence Pattern

Let's list the groups and their corresponding alphabetical positions:

• Group 1: $PIK$ corresponds to positions (16, 9, 11)
• Group 2: $IBD$ corresponds to positions (9, 2, 4)
• Group 3: $BUW$ corresponds to positions (2, 21, 23)
• Group 4: $UNP$ corresponds to positions (21, 14, 16)

Now, let's find the difference between the positions of the corresponding letters in consecutive groups.

Step-by-Step Pattern Calculation

First Letter Progression:

The first letters are P (16), I (9), B (2), U (21).

• From P (16) to I (9): The change is $9 - 16 = -7$.
• From I (9) to B (2): The change is $2 - 9 = -7$.
• From B (2) to U (21): When moving backward from B, we wrap around the alphabet. The change is $2 - 7 = -5$. In a 26-letter alphabet, position $-5$ is equivalent to $26 - 5 = 21$, which is U. So, the difference is $-7$.

The pattern for the first letter is consistently subtracting 7 from its alphabetical position.

To find the next first letter, we apply this rule to U (21):

Calculation: $21 - 7 = 14$. The 14th letter is N.

Second Letter Progression:

The second letters are I (9), B (2), U (21), N (14).

• From I (9) to B (2): The change is $2 - 9 = -7$.
• From B (2) to U (21): As seen before, $2 - 7$ wraps around to position 21 (U). The difference is $-7$.
• From U (21) to N (14): The change is $14 - 21 = -7$.

The pattern for the second letter is also consistently subtracting 7.

To find the next second letter, we apply this rule to N (14):

Calculation: $14 - 7 = 7$. The 7th letter is G.

Third Letter Progression:

The third letters are K (11), D (4), W (23), P (16).

• From K (11) to D (4): The change is $4 - 11 = -7$.
• From D (4) to W (23): Moving backward from D, $4 - 7 = -3$. This wraps around to position $26 - 3 = 23$, which is W. The difference is $-7$.
• From W (23) to P (16): The change is $16 - 23 = -7$.

The pattern for the third letter is also consistently subtracting 7.

To find the next third letter, we apply this rule to P (16):

Calculation: $16 - 7 = 9$. The 9th letter is I.

Determining the Missing Group

By applying the consistent pattern of subtracting 7 positions from each letter of the previous group, we find that the next group is formed by the letters N, G, and I.

Therefore, the group that should come in place of the question mark (?) is NGI.