Finding the Missing Number in the Sequence

The question asks us to identify the pattern connecting the numbers in each row and use it to find the missing number represented by the question mark (?) in the second row. We are given three rows of numbers:

Row 1: 5, 6, 2, 4, 81

Row 2: 1, 3, 2, 4, ?

Row 3: 2, 3, 1, 5, 39

Each row seems to follow a specific rule where the first four numbers are used to calculate the fifth number.

Analyzing the Number Pattern in Rows

Let's analyze the relationship between the first four numbers and the fifth number in the given complete rows (Row 1 and Row 3) to find the hidden pattern.

Pattern Analysis in Row 1: 5, 6, 2, 4 → 81

We need to find a mathematical operation or combination of operations involving 5, 6, 2, and 4 that results in 81. Let's try some common approaches:

• Sum of the numbers: $5 + 6 + 2 + 4 = 17$. This is not 81.
• Product of the numbers: $5 \times 6 \times 2 \times 4 = 240$. This is not 81.
• Let's consider powers. We notice 81 is $9^2$ or $3^4$. Can we get 9 or 3 using the first four numbers? $5+4=9$. Can we get the power (2 or 4) using the remaining numbers (6, 2)? If the pattern is $(1st + 4th)^{\text{power}}$, then $(5+4)^\text{power} = 9^\text{power} = 81$. If the power is 2, $9^2 = 81$. Can we get 2 from 6 and 2? Yes, by using 2 directly. Let's hypothesize the pattern is $(1st + 4th)^{3rd}$. So $(5+4)^2 = 9^2 = 81$. This seems promising for Row 1.

Pattern Analysis in Row 3: 2, 3, 1, 5 → 39

Let's test the hypothesized pattern $(1st + 4th)^{3rd}$ on Row 3 (2, 3, 1, 5). According to this pattern, the fifth number should be $(2 + 5)^1 = 7^1 = 7$. However, the fifth number in Row 3 is 39. So, the pattern $(1st + 4th)^{3rd}$ is incorrect.

Let's try another pattern for Row 1 (5, 6, 2, 4 → 81).

• Consider the squares of the numbers: $5^2 = 25$, $6^2 = 36$, $2^2 = 4$, $4^2 = 16$.
• Let's try summing the squares: $25 + 36 + 4 + 16 = 61 + 4 + 16 = 65 + 16 = 81$. This matches the fifth number in Row 1!

Verifying the Pattern with Row 3: 2, 3, 1, 5 → 39

Let's test if the pattern "Sum of the squares of the first four numbers" works for Row 3 (2, 3, 1, 5 → 39):

• Calculate the squares: $2^2 = 4$, $3^2 = 9$, $1^2 = 1$, $5^2 = 25$.
• Sum the squares: $4 + 9 + 1 + 25 = 13 + 1 + 25 = 14 + 25 = 39$. This also matches the fifth number in Row 3!

The pattern is confirmed: The fifth number in each row is the sum of the squares of the first four numbers.

Finding the Missing Number in Row 2

Now we apply the confirmed pattern to Row 2: 1, 3, 2, 4, ?

The missing number is the sum of the squares of 1, 3, 2, and 4.

Calculation:

• Square of the first number: $1^2 = 1$
• Square of the second number: $3^2 = 9$
• Square of the third number: $2^2 = 4$
• Square of the fourth number: $4^2 = 16$
• Sum of the squares: $1 + 9 + 4 + 16 = 10 + 4 + 16 = 14 + 16 = 30$

The missing number in Row 2 is 30.

Revision Table: Number Pattern Analysis

Additional Information: Number Sequence Patterns

Number sequence or pattern questions are common in logical reasoning and aptitude tests. They require you to identify the underlying rule that generates the sequence or relates elements within a set.

Common types of number patterns include:

• Arithmetic sequences: Each term is obtained by adding a constant value to the previous term (e.g., 2, 4, 6, 8...).
• Geometric sequences: Each term is obtained by multiplying the previous term by a constant value (e.g., 3, 6, 12, 24...).
• Fibonacci sequence: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Square or Cube patterns: Terms are squares or cubes of natural numbers (e.g., $1^2, 2^2, 3^2, ...$ or $1^3, 2^3, 3^3, ...$).
• Prime number sequences: Sequences involving prime numbers (e.g., 2, 3, 5, 7, 11...).
• Combined operations: Patterns involving a mix of arithmetic operations, powers, or other functions applied to previous terms or position numbers. The pattern in this question, summing the squares of previous numbers in the row, is an example of a combined operation pattern applied across elements within a set.

Solving these problems often involves checking for simple patterns first (addition, subtraction, multiplication, division) and then exploring more complex relationships like squares, cubes, differences between terms, or relationships between different elements in a set, as demonstrated in this problem.