Completing the Given Letter Series

The problem asks us to find the sequence of letters that will complete the given series:

\(l m \_ o \_ n \_ l n o \_ \_ \_ l m n\)

There are 6 blanks in the series. We need to examine the options to find the correct sequence to fill these blanks and identify a pattern.

Analyzing the Series with the Correct Option

Let's take the sequence from the correct option, which is 'nmolmo', and place these letters into the blanks in order:

Original Series: \(l m \_ o \_ n \_ l n o \_ \_ \_ l m n\)

Inserting 'nmolmo': \(l m \textbf{n} o \textbf{m} n \textbf{o} l n o \textbf{l} \textbf{m} \textbf{o} l m n\)

Identifying the Pattern in the Completed Series

Let's look at the completed series: \(l m n o m n o l n o l m o l m n\). We can try to find a repeating pattern or a rule governing the sequence.

Observing the letters involved (l, m, n, o) and the length of the series (16 characters), let's try grouping the letters into blocks.

If we group the series into blocks of 4 characters, we get:

• Block 1: \(l m n o\)
• Block 2: \(m n o l\)
• Block 3: \(n o l m\)
• Block 4: \(o l m n\)

We can see a clear pattern here. Each block of 4 characters is a cyclic shift of the previous block. Starting with 'l m n o', the next block starts with the second letter ('m') and follows the same order, cycling 'l' to the end. This continues for the subsequent blocks.

Verifying the Pattern against the Original Blanks

Now let's see if filling the blanks according to this pattern matches the correct option 'nmolmo'. The original series has blanks at positions 3, 5, 7, 11, 12, and 13.

Using the pattern:

• Block 1 (\(l m n o\)) fills positions 1-4: \(l m \textbf{n} o\). The character at position 3 is 'n'.
• Block 2 (\(m n o l\)) fills positions 5-8: \(\textbf{m} n \textbf{o} l\). The character at position 5 is 'm', and position 7 is 'o'.
• Block 3 (\(n o l m\)) fills positions 9-12: \(n o \textbf{l} \textbf{m}\). The character at position 11 is 'l', and position 12 is 'm'.
• Block 4 (\(o l m n\)) fills positions 13-16: \(\textbf{o} l m n\). The character at position 13 is 'o'.

The characters filled in the blanks are, in order: 'n', 'm', 'o', 'l', 'm', 'o'.

Conclusion

The sequence of characters derived from the pattern that fills the blanks is 'nmolmo'. This matches the sequence provided in the correct option. Therefore, the pattern of cyclic shifts of the block 'l m n o' correctly completes the given series.