Understanding Letter and Number Analogy Patterns

This question asks us to identify the relationship between the terms in the given pairs and apply the same relationship to find the missing term. The given pairs are H64 : J100 and L144 : N196. We need to find the term related to U441 in the same way.

Let's break down the pattern into two parts: the letter pattern and the number pattern.

Analyzing the Letter Pattern

Let's look at the relationship between the first letters of each pair:

• H to J
• L to N
• U to ?

We can determine the position of each letter in the English alphabet:

• H is the 8th letter. J is the 10th letter. The difference is $10 - 8 = +2$.
• L is the 12th letter. N is the 14th letter. The difference is $14 - 12 = +2$.

It is clear that the letter in the second term of each pair is two positions ahead of the letter in the first term.

Applying this pattern to the third pair:

• U is the 21st letter. The letter related to U should be $21 + 2 = 23$rd letter.
• The 23rd letter is W.

So, the letter for the missing term is W.

Analyzing the Number Pattern

Now let's look at the numbers in each pair:

• 64 and 100
• 144 and 196
• 441 and ?

Let's see how these numbers relate to the letters or their positions:

• For H64 : J100: The letters are H (8th) and J (10th). Notice that $8^2 = 64$ and $10^2 = 100$. The numbers are the squares of the alphabetical positions of the letters they are paired with.
• For L144 : N196: The letters are L (12th) and N (14th). Notice that $12^2 = 144$ and $14^2 = 196$. This pattern holds true for the second pair as well.

The pattern for the number seems to be the square of the alphabetical position of the letter it is associated with.

Applying this pattern to the third pair:

• The first term is U441. U is the 21st letter, and $21^2 = 441$. This confirms the pattern for the first part of the pair.
• The second term's letter is W (as determined from the letter pattern analysis). W is the 23rd letter.
• The number associated with W should be the square of its alphabetical position, which is $23^2$.

Calculating the Number

We need to calculate $23^2$:

\begin{equation*} 23^2 = 23 \times 23 \end{equation*}

\begin{equation*} 23 \times 23 = 529 \end{equation*}

So, the number for the missing term is 529.

Combining Letter and Number Patterns

Combining the results from the letter and number pattern analysis:

• The letter is W.
• The number is 529.

Therefore, the missing term is W529.

Summary of the Pattern

The complete pattern is: The second term is derived from the first by advancing the letter by two positions in the alphabet and taking the square of the new letter's alphabetical position as the number.

Verifying with Options

The derived term is W529. Let's check the given options:

• Option 1: W582 - Incorrect (Number is wrong)
• Option 2: W529 - Correct (Matches our derived term)
• Option 3: V484 - Incorrect (Letter and number are wrong)
• Option 4: V221 - Incorrect (Letter and number are wrong)

The option that matches our result is W529.

Revision Table: Key Concepts

Additional Information: Solving Analogy Problems

Analogy problems often test your ability to find relationships between pairs of items. These relationships can be based on various rules, including:

• Alphabetical Position: Using the position of letters (A=1, B=2, ... Z=26).
• Skipping Letters: Adding or subtracting a fixed number of positions.
• Number Series: Patterns involving squares, cubes, prime numbers, addition, subtraction, multiplication, or division.
• Combination of Patterns: Combining letter and number patterns, sometimes involving operations on the position or other characteristics.

To solve such problems effectively:

1. Observe the first pair closely to identify the relationship.
2. Check if the same relationship applies to the second pair.
3. If the pattern is consistent, apply it to the third item to find the missing fourth item.
4. Always verify your answer by checking the options provided.