Solving Letter Series Questions: Finding the Missing Term

This question asks us to find the missing term in the given letter series: EBA, _______, UJI, ONO, IRU.

To solve letter series problems, we typically analyze the pattern in the position of the letters in the alphabet. Let's assign a number to each letter based on its position (A=1, B=2, ..., Z=26).

The given terms are:

• EBA: E is the 5th letter, B is the 2nd, A is the 1st. (5, 2, 1)
• UJI: U is the 21st letter, J is the 10th, I is the 9th. (21, 10, 9)
• ONO: O is the 15th letter, N is the 14th, O is the 15th. (15, 14, 15)
• IRU: I is the 9th letter, R is the 18th, U is the 21st. (9, 18, 21)

The series in terms of letter positions is: (5, 2, 1), _______, (21, 10, 9), (15, 14, 15), (9, 18, 21).

Let's analyze the pattern for each position (first, second, and third letter) separately.

Pattern for the First Letter

The sequence of the first letters is E, ?, U, O, I. Their positions are 5, ?, 21, 15, 9.

Let's look at the differences between consecutive known terms from the end:

• O(15) to I(9): \(9 - 15 = -6\). This means we subtract 6 from the position.
• U(21) to O(15): \(15 - 21 = -6\). This also means we subtract 6 from the position.

It appears there is a consistent subtraction of 6 for the last two steps. Let's see if this pattern extends backwards.

• Assuming the step before U was also subtracting 6: The letter before U would be at position \(21 - 6 = 15\), which is O. But the term before UJI is the missing one, and before the missing one is E.

Let's examine the differences starting from the first known term, E(5), towards the missing term (let's call its first letter position \(x\)), then to U(21).

• E(5) to ?: Let the change be \(d_1\). So, \(x = 5 + d_1\).
• ? to U(21): Let the change be \(d_2\). So, \(21 = x + d_2\).

From U(21) to O(15) the change is -6. From O(15) to I(9) the change is -6.

The pattern of changes might be \((d_1, d_2, -6, -6)\). Let's check the options provided for the missing term. The correct option is given as AFE. Let's check the letters in AFE:

• AFE: A is 1st, F is 6th, E is 5th. (1, 6, 5)

If AFE is the missing term, the sequence of first letters and their positions is E(5), A(1), U(21), O(15), I(9).

Let's look at the differences between consecutive terms now:

• E(5) to A(1): \(1 - 5 = -4\). We subtract 4.
• A(1) to U(21): \(21 - 1 = 20\). We add 20. (Or subtract 6 modulo 26: \(1 - 6 = -5 \equiv 21 \pmod{26}\)). Subtracting 6 seems more likely given the subsequent steps.
• U(21) to O(15): \(15 - 21 = -6\). We subtract 6.
• O(15) to I(9): \(9 - 15 = -6\). We subtract 6.

The pattern for the first letter is to subtract 4, then subtract 6 repeatedly (modulo 26). E \(\xrightarrow{-4}\) A \(\xrightarrow{-6}\) U \(\xrightarrow{-6}\) O \(\xrightarrow{-6}\) I.

Pattern for the Second Letter

The sequence of the second letters is B, ?, J, N, R. Their positions are 2, ?, 10, 14, 18.

Let's look at the differences between consecutive known terms:

• J(10) to N(14): \(14 - 10 = 4\). We add 4.
• N(14) to R(18): \(18 - 14 = 4\). We add 4.

It appears there is a consistent addition of 4. Let's test this pattern backward and forward from the known terms:

• Let the missing second letter position be \(y\). B(2) \(\xrightarrow{+4}\) \(y\) \(\xrightarrow{+4}\) J(10) \(\xrightarrow{+4}\) N(14) \(\xrightarrow{+4}\) R(18).
• From B(2) add 4: \(2 + 4 = 6\). The 6th letter is F. So, \(y = 6\).
• Check if adding 4 to this gives J: \(6 + 4 = 10\). The 10th letter is J. This matches the series.

The pattern for the second letter is adding 4 consistently. The sequence is B(2), F(6), J(10), N(14), R(18).

Pattern for the Third Letter

The sequence of the third letters is A, ?, I, O, U. Their positions are 1, ?, 9, 15, 21.

Let's look at the differences between consecutive known terms from the end:

• O(15) to U(21): \(21 - 15 = 6\). We add 6.
• I(9) to O(15): \(15 - 9 = 6\). We add 6.

It appears there is a consistent addition of 6 for the last two steps. Let's test this pattern backward from the known terms:

• Let the missing third letter position be \(z\). A(1) \(\xrightarrow{d_3}\) \(z\) \(\xrightarrow{d_4}\) I(9) \(\xrightarrow{+6}\) O(15) \(\xrightarrow{+6}\) U(21).
• From I(9) subtract 6: \(9 - 6 = 3\). The 3rd letter is C. If \(d_4 = 6\), then \(z=3\).
• From A(1) add \(d_3\): \(z = 1 + d_3\). If \(z=3\), then \(d_3 = 2\).

This would suggest the differences are 2, 6, 6, 6. However, let's use the sequence implied by the missing term AFE, whose third letter is E(5).

If E(5) is the missing third letter, the sequence is A(1), E(5), I(9), O(15), U(21).

Let's look at the differences now:

• A(1) to E(5): \(5 - 1 = 4\). We add 4.
• E(5) to I(9): \(9 - 5 = 4\). We add 4.
• I(9) to O(15): \(15 - 9 = 6\). We add 6.
• O(15) to U(21): \(21 - 15 = 6\). We add 6.

The pattern for the third letter is adding 4 twice, then adding 6 twice. The sequence is A(1) \(\xrightarrow{+4}\) E(5) \(\xrightarrow{+4}\) I(9) \(\xrightarrow{+6}\) O(15) \(\xrightarrow{+6}\) U(21).

Combining the Patterns

Based on the analysis of each letter's position pattern:

• The first letter follows the pattern of subtracting 4, then subtracting 6 repeatedly. Starting with E, the next letter is A.
• The second letter follows the pattern of adding 4 consistently. Starting with B, the next letter is F.
• The third letter follows the pattern of adding 4 twice, then adding 6 twice. Starting with A, the next letter is E.

Combining these letters, the missing term is AFE.

Verification of the Missing Term

Let's write out the series with AFE included and check the patterns:

EBA \(\xrightarrow{\text{Pattern } 1}\) AFE \(\xrightarrow{\text{Pattern } 2}\) UJI \(\xrightarrow{\text{Pattern } 3}\) ONO \(\xrightarrow{\text{Pattern } 4}\) IRU

The patterns hold true for all steps when AFE is inserted as the missing term.

Therefore, the option that fills the blank and completes the series is AFE.

Revision Table: Summary of Patterns

Additional Information: Types of Letter Series Patterns

Letter series questions are common in logical reasoning tests. They can have various types of patterns, such as:

• Arithmetic Progression: Each letter's position changes by a constant value. (Like the second letter pattern in this problem).
• Multiple Arithmetic Progressions: Different patterns for different letter positions, or alternating patterns between terms. (Like the patterns for the first and third letters here).
• Alphabetic Order: Letters skip a fixed number of places or follow the natural order.
• Reverse Order: Letters follow the reverse order of the alphabet.
• Vowel/Consonant Patterns: The series might involve sequences of only vowels or consonants, or patterns related to their properties.
• Sum/Difference Patterns: The positions of letters within a term or between terms might follow arithmetic or other numerical patterns.
• Alternating Patterns: Different patterns might apply to alternate terms in the series.

Solving these questions requires careful observation, knowledge of alphabet positions, and testing different potential patterns.