From the series 7, 18, 40, 106, 183, 282, 403, one term is inconsistent with the underlying numeric pattern. Which number is the wrong term?

Introduction / Context:
This question asks us to identify an incorrect term in an increasing number sequence. The numbers grow rapidly, which suggests that the pattern is in the differences or in a second level series rather than a simple linear progression. Detecting such higher order patterns is a key skill in advanced number series problems.

Given Data / Assumptions:

• The series given is: 7, 18, 40, 106, 183, 282, 403.
• Exactly one term is wrong and must be identified.
• We assume that the underlying correct sequence follows a reasonably simple rule, likely in the differences between terms.
• The differences themselves may have a recognizable pattern, for example multiples of a constant.

Concept / Approach:
To find the wrong term, we compare consecutive differences first. If these differences do not immediately show a clear structure, we examine whether they are all multiples of some number or follow a simple pattern with one anomaly. In this sequence, the differences turn out to be multiples of 11, hinting at a deeper pattern in which one of the increments is inconsistent with the rest and therefore points to the incorrect term.

Step-by-Step Solution:
Step 1: Compute differences between consecutive terms. 18 - 7 = 11. 40 - 18 = 22. 106 - 40 = 66. 183 - 106 = 77. 282 - 183 = 99. 403 - 282 = 121. Step 2: Note that these differences are 11, 22, 66, 77, 99, 121. Step 3: Factor each difference by 11: 11*(1, 2, 6, 7, 9, 11). Step 4: Suspect a simpler intended pattern such as using consecutive odd multiples of 11: 11, 33, 55, 77, 99, 121 (i.e., 11*1, 11*3, 11*5, 11*7, 11*9, 11*11). Step 5: Reconstruct the series using these intended differences, starting from 7. Next term: 7 + 11 = 18 (matches). Next term: 18 + 33 = 51 (should be here). Next term: 51 + 55 = 106 (matches 106). Then: 106 + 77 = 183; 183 + 99 = 282; 282 + 121 = 403, all matching the given sequence. Step 6: The only mismatch is at the position where 51 should appear; the given term is 40. Step 7: Therefore, 40 is the wrong term in the series.

Verification / Alternative check:
The corrected series is 7, 18, 51, 106, 183, 282, 403. Its differences are 11, 33, 55, 77, 99, 121, exactly 11 times the odd numbers 1, 3, 5, 7, 9, 11. This pattern is elegant, symmetric and fully consistent. The fact that only one given term (40) disrupts this otherwise perfect progression confirms it as the erroneous entry.

Why Other Options Are Wrong:
Removing 18, 106, 183 or 282 would destroy the smooth sequence of differences that are all odd multiples of 11. With those terms in place, the increments remain neatly patterned once we correct the third term to 51. Therefore, none of the other options can be considered the unique wrong term.

Common Pitfalls:
It is easy to be misled by the large jumps (for example 40 to 106 and 106 to 183) and give up on finding a simple pattern. Some students may also attempt to fit a complex formula for the nth term instead of exploiting the structure in the differences. The key insight here is to notice that all differences are near multiples of 11 and then refine the idea to odd multiples of 11, which exposes the single anomaly cleanly.

Final Answer:
The term that breaks the intended pattern of odd multiples of 11 as differences is 40.