Answer :
To solve this problem, we need to determine which number in the series [tex]$2, 6, 15, 31, 56, 93$[/tex] is incorrect based on a pattern.
Here's a step-by-step approach:
1. Identify a pattern:
Let's calculate the differences between consecutive terms in the series to see if they follow a recognizable pattern.
- Difference between [tex]$6$[/tex] and [tex]$2$[/tex]: [tex]$6 - 2 = 4$[/tex]
- Difference between [tex]$15$[/tex] and [tex]$6$[/tex]: [tex]$15 - 6 = 9$[/tex]
- Difference between [tex]$31$[/tex] and [tex]$15$[/tex]: [tex]$31 - 15 = 16$[/tex]
- Difference between [tex]$56$[/tex] and [tex]$31$[/tex]: [tex]$56 - 31 = 25$[/tex]
- Difference between [tex]$93$[/tex] and [tex]$56$[/tex]: [tex]$93 - 56 = 37$[/tex]
2. Recognize the pattern of differences:
The differences we calculated are [tex]$4, 9, 16, 25, 37$[/tex]. Notice that [tex]$4, 9, 16, 25$[/tex] are perfect squares: [tex]$2^2, 3^2, 4^2, 5^2$[/tex]. The expectation for the next number following this pattern would be [tex]$6^2 = 36$[/tex]. However, the difference we have is [tex]$37$[/tex].
3. Identify the incorrect number:
The last difference being [tex]$37$[/tex] suggests that the number [tex]$93$[/tex] is incorrect because we expected a difference of [tex]$36$[/tex] from the pattern. Therefore, the number that does not fit the expected pattern is [tex]$93$[/tex].
So, the incorrect number in the series is 93.
Here's a step-by-step approach:
1. Identify a pattern:
Let's calculate the differences between consecutive terms in the series to see if they follow a recognizable pattern.
- Difference between [tex]$6$[/tex] and [tex]$2$[/tex]: [tex]$6 - 2 = 4$[/tex]
- Difference between [tex]$15$[/tex] and [tex]$6$[/tex]: [tex]$15 - 6 = 9$[/tex]
- Difference between [tex]$31$[/tex] and [tex]$15$[/tex]: [tex]$31 - 15 = 16$[/tex]
- Difference between [tex]$56$[/tex] and [tex]$31$[/tex]: [tex]$56 - 31 = 25$[/tex]
- Difference between [tex]$93$[/tex] and [tex]$56$[/tex]: [tex]$93 - 56 = 37$[/tex]
2. Recognize the pattern of differences:
The differences we calculated are [tex]$4, 9, 16, 25, 37$[/tex]. Notice that [tex]$4, 9, 16, 25$[/tex] are perfect squares: [tex]$2^2, 3^2, 4^2, 5^2$[/tex]. The expectation for the next number following this pattern would be [tex]$6^2 = 36$[/tex]. However, the difference we have is [tex]$37$[/tex].
3. Identify the incorrect number:
The last difference being [tex]$37$[/tex] suggests that the number [tex]$93$[/tex] is incorrect because we expected a difference of [tex]$36$[/tex] from the pattern. Therefore, the number that does not fit the expected pattern is [tex]$93$[/tex].
So, the incorrect number in the series is 93.
