Understanding the Number Pattern Question

This question asks us to analyze a given pattern of numbers and find the number that logically replaces the question mark. Such questions test our ability to identify the underlying rule or relationship between the numbers in the pattern.

The pattern is presented in three rows:

We need to find a consistent rule that connects the first two numbers in each row to the third number.

Analyzing the Number Pattern Logic

Let's look closely at the numbers in each row. We have some relatively large numbers and some smaller numbers. Some numbers seem to be perfect cubes or perfect squares.

• In Row 1, we have 125 and 25. We know that $125 = 5^3$ and $25 = 5^2$. The third number is 10.
• In Row 2, we have 216 and 49. We know that $216 = 6^3$ and $49 = 7^2$. The third number is 13.
• In Row 3, we have 27 and 121. We know that $27 = 3^3$ and $121 = 11^2$. The third number is ?.

Let's try to find a relationship between the bases of the powers or the numbers themselves using mathematical operations.

Discovering the Pattern Rule

Consider the cube root of the first number and the square root of the second number in each row.

• For Row 1: The first number is 125, its cube root is $\sqrt[3]{125} = 5$. The second number is 25, its square root is $\sqrt{25} = 5$.
• For Row 2: The first number is 216, its cube root is $\sqrt[3]{216} = 6$. The second number is 49, its square root is $\sqrt{49} = 7$.
• For Row 3: The first number is 27, its cube root is $\sqrt[3]{27} = 3$. The second number is 121, its square root is $\sqrt{121} = 11$.

Now let's see if adding or combining these roots gives us the third number in the row:

• Row 1: $\sqrt[3]{125} + \sqrt{25} = 5 + 5 = 10$. This matches the third number in Row 1.
• Row 2: $\sqrt[3]{216} + \sqrt{49} = 6 + 7 = 13$. This matches the third number in Row 2.

It appears the pattern is: Cube root of the first number + Square root of the second number = Third number.

Calculating the Missing Number

Let's apply this rule to the third row to find the missing number.

• First number in Row 3 is 27. Its cube root is $\sqrt[3]{27} = 3$.
• Second number in Row 3 is 121. Its square root is $\sqrt{121} = 11$.

According to the pattern rule, the missing number is the sum of these two values:

Missing number = $\sqrt[3]{27} + \sqrt{121} = 3 + 11 = 14$.

Final Answer for the Number Pattern

The number that replaces the question mark (?) in the pattern is 14.

Revision Table: Key Concepts

Additional Information: Solving Number Pattern Questions

Solving number pattern questions often involves looking for mathematical relationships between the numbers. Here are some common things to check:

• Arithmetic Operations: Addition, subtraction, multiplication, division between adjacent numbers or numbers in corresponding positions.
• Powers and Roots: Numbers might be squares, cubes, square roots, cube roots, etc., of other numbers in the pattern.
• Differences/Ratios: Look at the difference or ratio between consecutive numbers. These differences/ratios might form a new pattern.
• Combinations: The relationship might involve a combination of operations (e.g., multiply and then add).
• Position/Index: The number might relate to its position in the sequence (e.g., $n^2$, $2n+1$).
• Digits: Sometimes, the pattern relates to the sum or product of the digits of the numbers.

Always test your hypothesized rule on all given parts of the pattern before applying it to find the missing element.