Difficulty: Medium
Correct Answer: 34
Explanation:
Introduction / Context:This is a decreasing number series: 102, 85, 68, 51, ?. Here, each term is smaller than the previous one by a constant amount, and these amounts themselves form a simple pattern. Such questions test the ability to identify regular changes in differences between terms.
Given Data / Assumptions:
Concept / Approach:For a decreasing series, we compute differences between consecutive terms. If these differences increase or decrease in a clear pattern, we extend this pattern to find the missing term. Here, the differences turn out to be equal, which means the series is an arithmetic progression with a constant negative common difference.
Step-by-Step Solution:Step 1: Compute first differences.85 - 102 = -17.68 - 85 = -17.51 - 68 = -17.Step 2: Observe the pattern.Each term is 17 less than the previous term. So the common difference is -17.Step 3: Apply the same difference for the missing term.Missing term = 51 - 17 = 34.
Verification / Alternative check:Write the full series including the missing term: 102, 85, 68, 51, 34. Now recompute differences: -17, -17, -17, -17. The pattern of a constant negative difference holds perfectly for all consecutive pairs, confirming that 34 is correct.
Why Other Options Are Wrong:40: Difference from 51 would be -11, which breaks the uniform difference of -17.37: Difference from 51 would be -14, not consistent with other steps.31: Difference from 51 would be -20, again inconsistent with the established pattern.
Common Pitfalls:Because the starting number 102 is relatively large, some candidates look for more complex multiplicative patterns. However, when a series decreases smoothly, checking simple differences is usually the fastest and most reliable method. Do not overcomplicate the series before verifying whether it is just an arithmetic progression.
Final Answer:The missing number in the arithmetic series with common difference -17 is 34.
Discussion & Comments