Number Set Relationship Analysis

The question presents three sets of numbers: (2, 8, 100), (7, 2, 81), and (4, 7, ?). The goal is to identify the relationship or pattern connecting the first two numbers (let's call them a and b) to the third number (c) in the first two sets and then apply this pattern to find the missing number in the third set.

Identifying the Pattern

Let's examine the first set: (2, 8, 100).

• We need to find a connection between 2 and 8 that results in 100.
• Consider adding the first two numbers: $2 + 8 = 10$.
• Now, let's see if squaring this sum relates to the third number: $10^2 = 10 \times 10 = 100$.
• This matches the third number in the first set.

Now, let's test this potential pattern using the second set: (7, 2, 81).

• Add the first two numbers: $7 + 2 = 9$.
• Square the sum: $9^2 = 9 \times 9 = 81$.
• This also matches the third number in the second set.

The pattern is confirmed: The third number in each set is the square of the sum of the first two numbers. Mathematically, for a set $(a, b, c)$, the relationship is $c = (a + b)^2$.

Calculating the Missing Number

We apply the identified pattern to the third set: (4, 7, ?).

• Here, $a = 4$ and $b = 7$.
• First, find the sum of the first two numbers: $a + b = 4 + 7 = 11$.
• Next, square the sum: $(a + b)^2 = 11^2$.
• Calculate the final value: $11^2 = 11 \times 11 = 121$.

Therefore, the missing number (?) in the third set is 121.

Conclusion

Based on the consistent mathematical relationship $c = (a + b)^2$ observed in the first two sets, the missing number in the third set (4, 7, ?) is 121.