Understanding Letter Series Completion

This question asks us to find the missing term in a given series of letter groups. To solve this type of series completion problem, we need to identify the pattern or rule that connects consecutive terms. The pattern can be based on the position of the letters in the alphabet, mathematical operations applied to these positions, or a combination of rules for different letter positions within the group.

Analyzing the Given Letter Series

The given series is:

PMRF, PDZV, PUHL, PLPB, ?

Let's break down each term and look at the letters at each position (first, second, third, and fourth) across the series.

Pattern for the First Letter

The first letters of the terms are:

• P (in PMRF)
• P (in PDZV)
• P (in PUHL)
• P (in PLPB)

The first letter is consistently 'P' in all terms. So, the first letter of the missing term will also be 'P'.

Pattern for the Second Letter

The second letters of the terms are:

• M (Position 13)
• D (Position 4)
• U (Position 21)
• L (Position 12)
• ?

Let's look at the difference in their alphabetical positions:

• M (13) to D (4): The difference is $4 - 13 = -9$. Cyclically, this is $26 - 9 = 17$ positions forward. Or, $13 + 17 = 30 \equiv 4 \pmod{26}$.
• D (4) to U (21): The difference is $21 - 4 = 17$. So, $+17$.
• U (21) to L (12): The difference is $12 - 21 = -9$. Cyclically, this is $26 - 9 = 17$ positions forward. Or, $21 + 17 = 38 \equiv 12 \pmod{26}$.

The pattern for the second letter appears to be adding 17 to the alphabetical position (and wrapping around if the sum exceeds 26). Let's apply this to the last term's second letter, L (Position 12):

Position of L is 12. Next position = $12 + 17 = 29$. Since there are only 26 letters, we find the equivalent position cyclically: $29 - 26 = 3$.

The letter at position 3 is C. So, the second letter of the missing term is 'C'.

Pattern for the Third Letter

The third letters of the terms are:

• R (Position 18)
• Z (Position 26)
• H (Position 8)
• P (Position 16)
• ?

Let's look at the difference in their alphabetical positions:

• R (18) to Z (26): The difference is $26 - 18 = 8$. So, $+8$.
• Z (26) to H (8): The difference is $8 - 26 = -18$. Cyclically, this is $26 - 18 = 8$ positions forward. Or, $26 + 8 = 34 \equiv 8 \pmod{26}$. So, $+8$.
• H (8) to P (16): The difference is $16 - 8 = 8$. So, $+8$.

The pattern for the third letter appears to be adding 8 to the alphabetical position (and wrapping around). Let's apply this to the last term's third letter, P (Position 16):

Position of P is 16. Next position = $16 + 8 = 24$.

The letter at position 24 is X. So, the third letter of the missing term is 'X'.

Pattern for the Fourth Letter

The fourth letters of the terms are:

• F (Position 6)
• V (Position 22)
• L (Position 12)
• B (Position 2)
• ?

Let's look at the difference in their alphabetical positions:

• F (6) to V (22): The difference is $22 - 6 = 16$. So, $+16$.
• V (22) to L (12): The difference is $12 - 22 = -10$. Cyclically, this is $26 - 10 = 16$ positions forward. Or, $22 + 16 = 38 \equiv 12 \pmod{26}$. So, $+16$.
• L (12) to B (2): The difference is $2 - 12 = -10$. Cyclically, this is $26 - 10 = 16$ positions forward. Or, $12 + 16 = 28 \equiv 2 \pmod{26}$. So, $+16$.

The pattern for the fourth letter appears to be adding 16 to the alphabetical position (and wrapping around). Let's apply this to the last term's fourth letter, B (Position 2):

Position of B is 2. Next position = $2 + 16 = 18$.

The letter at position 18 is R. So, the fourth letter of the missing term is 'R'.

Combining the Patterns to Find the Missing Term

Based on our analysis of each position:

• First letter: P
• Second letter: C
• Third letter: X
• Fourth letter: R

Combining these letters gives us the next term in the series: PCXR.

Checking the Options

Let's compare our derived term with the given options:

• Option 1: PCMH
• Option 2: PHTD
• Option 3: PCXR
• Option 4: PQRS

Our derived term, PCXR, matches Option 3.

Conclusion

By analyzing the patterns in each letter position of the given series PMRF, PDZV, PUHL, PLPB, we determined the rules governing the changes. The first letter remains constant (P), the second letter advances by 17 positions (cyclically), the third letter advances by 8 positions (cyclically), and the fourth letter advances by 16 positions (cyclically). Applying these rules to the last term PLPB, we found the next term to be PCXR.

Revision Table: Letter Series Analysis

Understanding how to analyze letter series is crucial for reasoning questions. Here’s a quick recap of steps:

• List the terms clearly.
• Separate the letters by their position within each term.
• Analyze the sequence of letters for each position independently.
• Look for patterns: constant letters, arithmetic progression in letter positions, cyclic shifts (modulo 26), alternating patterns, etc.
• Apply the identified pattern(s) to the last term to find the missing term.
• Verify your answer with the given options.

Additional Information: Alphabetical Positions and Cyclical Shifts

For letter series problems, it's helpful to know the alphabetical position of each letter (A=1, B=2, ..., Z=26). Cyclical shifts mean that if you go past Z (position 26), you loop back to A (position 1). For example, adding 3 to Y (position 25): 25 + 3 = 28. On a 26-letter alphabet, position 28 is equivalent to position $28 - 26 = 2$, which is B. This is often represented using modulo arithmetic, i.e., $(Position + Shift) \pmod{26}$. If the result is 0, it corresponds to Z (position 26).

Example: Z (26) + 8 = 34. $34 \pmod{26} = 8$. Position 8 is H.

Example: B (2) + 16 = 18. $18 \pmod{26} = 18$. Position 18 is R.