Points to Remember:
- The series involves a pattern of increasing numbers.
- The pattern needs to be identified to find the missing number.
- The solution requires a mathematical approach.
Introduction:
This question requires identifying a numerical pattern within a given sequence: 3, 7, 15, 31, ?, 127, 255. The approach is purely factual and analytical, focusing on identifying the mathematical relationship between consecutive numbers in the series to determine the missing value. This type of question tests pattern recognition and mathematical reasoning skills.
Body:
Identifying the Pattern:
Let’s analyze the differences between consecutive numbers in the series:
- 7 – 3 = 4
- 15 – 7 = 8
- 31 – 15 = 16
- 127 – ? = x
- 255 – 127 = 128
Notice that the differences themselves form a pattern: 4, 8, 16… This suggests that each difference is double the previous difference. Therefore, the pattern is based on doubling the difference between consecutive numbers.
Determining the Missing Number:
Following the pattern, the next difference should be double the previous one (16 x 2 = 32). Therefore:
- 31 + 32 = 63
The missing number is 63.
Verifying the Pattern:
Let’s check if this fits the rest of the series:
- 127 – 63 = 64 (which is 32 x 2)
- 255 – 127 = 128 (which is 64 x 2)
The pattern holds true throughout the series.
Conclusion:
The missing number in the series 3, 7, 15, 31, ?, 127, 255 is 63. This was determined by analyzing the differences between consecutive numbers and identifying a pattern of doubling the difference at each step. The consistent application of this pattern verifies the accuracy of the solution. This exercise highlights the importance of pattern recognition and logical reasoning in solving mathematical problems. Further exploration of similar number series could enhance mathematical problem-solving skills.