Understanding the Number Sets

The question presents three sets of numbers. The first two sets follow a specific mathematical rule that relates the first two numbers to the third number. We need to identify this rule and apply it to the third set to find the missing number represented by '?'.

Analysing Set 1: (3, 6, 81)

Let the numbers in a set be represented as (a, b, c).

In the first set, we have a = 3, b = 6, and c = 81.

Let's explore potential relationships:

• Adding the first two numbers: $3 + 6 = 9$.
• Squaring the sum: $9^2 = 81$.

This suggests the relationship might be $ (a + b)^2 = c $. Let's test this rule with the second set.

Analysing Set 2: (2, 2, 16)

In the second set, we have a = 2, b = 2, and c = 16.

Applying the suspected rule:

• Adding the first two numbers: $2 + 2 = 4$.
• Squaring the sum: $4^2 = 16$.

The rule $ (a + b)^2 = c $ holds true for the second set as well.

Applying the Relationship to Set 3: (4, 5, ?)

Now, we apply the confirmed rule $ (a + b)^2 = c $ to the third set, where a = 4 and b = 5.

1. Step 1: Sum the first two numbers. $ a + b = 4 + 5 = 9 $
2. Step 2: Square the result from Step 1. $ (9)^2 = 81 $

Therefore, the missing number represented by '?' is 81.

Conclusion

By identifying the pattern where the square of the sum of the first two numbers equals the third number ($ (a + b)^2 = c $), we found that the missing number in the set (4, 5, ?) is 81.