Understanding Number Analogy Patterns

The question asks us to find the missing number in a sequence based on a relationship observed in the given pairs. The given relationship is presented as:

$$26 : 6 :: ? : 11 :: 37 : 7$$

This means the relationship between 26 and 6 is the same as the relationship between the missing number and 11, and also the same as the relationship between 37 and 7. We need to identify the pattern in the known pairs (26 and 6, 37 and 7) and apply it to the pair involving 11 to find the missing number.

Analyzing the Known Number Pairs

Let's look at the first pair: 26 and 6.

How can these two numbers be related? Let's consider simple arithmetic operations or powers involving the second number (6).

• $6 \times 4 = 24$, $24 + 2 = 26$.
• $6 \times 5 = 30$, $30 - 4 = 26$.
• Consider the square of 6: $6^2 = 36$. How to get 26 from 36? $36 - 10 = 26$.
• Consider the number before 6, which is 5. Square 5: $5^2 = 25$. How to get 26 from 25? $25 + 1 = 26$.

This last relationship seems interesting: the first number is obtained by taking the number one less than the second number, squaring it, and adding 1. Let's check if this pattern holds for the other known pair.

Now let's look at the third pair: 37 and 7.

According to the potential pattern, we take the number one less than the second number (7), which is $7-1=6$. Square this number: $6^2 = 36$. Add 1 to this result: $36 + 1 = 37$.

This matches the first number in the pair (37). So, the pattern seems to be consistent for both given pairs.

Identifying the Pattern or Rule

The pattern observed is that the first number in each pair is obtained by squaring the number that is one less than the second number and then adding 1. If we represent the first number as $F$ and the second number as $S$, the relationship is:

$$F = (S - 1)^2 + 1$$

Let's verify this rule with the given pairs:

• For the pair 26 and 6: $S=6$. $F = (6 - 1)^2 + 1 = 5^2 + 1 = 25 + 1 = 26$. (Correct)
• For the pair 37 and 7: $S=7$. $F = (7 - 1)^2 + 1 = 6^2 + 1 = 36 + 1 = 37$. (Correct)

Applying the Pattern to Find the Missing Number

Now we apply this established pattern to the pair involving the missing number: ? and 11.

Here, the second number $S$ is 11. Let the missing first number be $X$. Using the pattern $F = (S - 1)^2 + 1$, we have:

$$X = (11 - 1)^2 + 1$$ $$X = (10)^2 + 1$$ $$X = 100 + 1$$ $$X = 101$$

So, the missing number is 101.

Verifying with Options

Let's check the provided options to see if 101 is among them.

1. 101
2. 122
3. 82
4. 145

The calculated missing number, 101, matches the first option.

Therefore, the correct option related to the fourth number (11) in the same way as the other pairs is 101.

Revision Table: Number Analogy

Additional Information: Solving Number Analogy Questions

Number analogy questions test your ability to identify numerical relationships. These relationships can involve various mathematical operations, including:

• Addition, subtraction, multiplication, division.
• Squares, cubes, square roots, cube roots.
• Prime numbers, composite numbers, odd/even numbers.
• Sequences (arithmetic progression, geometric progression).
• Digit manipulation (sum of digits, product of digits).
• Combinations of operations (e.g., square and add).

To solve number analogy problems effectively:

• Look for patterns in the given pairs first.
• Consider simple relationships before complex ones.
• Test potential patterns rigorously on all given examples.
• Once a pattern is confirmed, apply it to the pair with the missing number.
• Check your answer against the options provided.

Practice is key to recognizing different types of patterns quickly.