Analyzing the Number Series Puzzle

The question asks us to identify the missing number in the given numerical series: 552, ?, 650, 702, 756. To find the missing term, we need to carefully examine the relationship between the existing numbers and determine the pattern.

Identifying the Pattern in the Series

Let's calculate the differences between the consecutive known terms to uncover the pattern:

• The difference between 756 and 702 is: $756 - 702 = 54$
• The difference between 702 and 650 is: $702 - 650 = 52$

We notice that the differences between the terms are not constant. Instead, they are increasing. The difference increased from 52 to 54. The increase in the difference is $54 - 52 = 2$. This suggests that the differences between consecutive terms form an arithmetic sequence, increasing by 2 each time.

Based on this observation, we can deduce the preceding differences:

• The difference between the term represented by '?' and 650 should be $52 - 2 = 50$.
• The difference between 552 and the term represented by '?' should be $50 - 2 = 48$.

Therefore, the sequence of differences between consecutive terms is 48, 50, 52, 54.

Step-by-Step Solution

Let the missing number be represented by '$x$'. The series is 552, $x$, 650, 702, 756.

1. Using the first difference: The difference between the first term (552) and the second term ($x$) should be 48. $$x - 552 = 48$$ To find $x$, we add 552 and 48: $$x = 552 + 48$$ $$x = 600$$
2. Verifying with the second difference: Now, let's check if this value of $x$ fits the pattern with the next term. The difference between the second term ($x$) and the third term (650) should be 50. $$650 - x = 50$$ Substitute $x = 600$: $$650 - 600 = 50$$ This calculation confirms that the difference is indeed 50, matching our deduced pattern.
3. Confirming the rest of the series: The difference between 650 and 702 is $702 - 650 = 52$. The difference between 702 and 756 is $756 - 702 = 54$. These differences (52 and 54) align perfectly with the established pattern of increasing differences (48, 50, 52, 54).

The value $x = 600$ consistently satisfies the pattern identified in the series.

Conclusion

The number that correctly fits into the sequence 552, ?, 650, 702, 756, following the pattern of differences increasing by 2, is 600.