Choose the correct alternative that will continue the same pattern and replace the missing term x in the series 525, 430, 345, 270, x.

Introduction / Context:

This question involves a decreasing number series where each successive term is obtained by subtracting a regular, structured amount from the previous term. Such problems evaluate a candidate understanding of arithmetic progressions in the differences of a series. Instead of working directly with the original numbers, the focus is usually on the pattern formed by the differences between consecutive terms. Once that difference pattern is recognized, finding the missing term becomes straightforward.

Given Data / Assumptions:

• The given series is 525, 430, 345, 270, x.
• We must determine the single missing term x using the same pattern observed from the earlier consecutive terms.
• The numbers are decreasing, which suggests subtraction with a pattern in the subtracted values.
• Only one option will satisfy the same rule throughout the entire series.

Concept / Approach:

The most natural approach here is to look at consecutive differences. When we subtract each term from the previous one, we often find that the differences themselves either form an arithmetic progression, a geometric progression, or follow some simple rule such as constant changes. After establishing the pattern in these differences, we can apply the same logic to compute the missing term x. This method is widely used in number series questions, especially when the original numbers are moderately large and clearly not squares or cubes of a simple sequence.

Step-by-Step Solution:

Step 1: Compute the first difference: 525 - 430 = 95. Step 2: Compute the second difference: 430 - 345 = 85. Step 3: Compute the third difference: 345 - 270 = 75. Step 4: Observe that the differences form the descending sequence 95, 85, 75, which decreases by 10 each time. Step 5: Continue this pattern: the next difference should be 65, so x = 270 - 65 = 205.

Verification / Alternative check:

Verify the full pattern including the found value. Starting with 525, subtract 95 to get 430. From 430, subtract 85 to get 345. From 345, subtract 75 to get 270. From 270, subtract 65 to get 205. The differences now are 95, 85, 75, 65, forming a clean arithmetic progression with common difference minus 10. There is no break or irregularity in this pattern, demonstrating that the discovered value 205 is fully consistent with the structure of the given series.

Why Other Options Are Wrong:

Option 255 would produce a difference of only 15 from 270, which completely breaks the sequence of differences. Option 235 gives a difference of 35 and again does not follow the minus 10 pattern. Option 175 gives a difference of 95, which would repeat the first difference and disrupt the steadily decreasing nature. Option 195 yields a difference of 75, repeating an earlier difference and failing to maintain the arithmetic progression. Therefore only 205 preserves the descending difference pattern of 95, 85, 75, 65.

Common Pitfalls:

Students often try to relate each term to the very first term using complex formulas instead of simply examining consecutive differences. Another common mistake is to assume multiplication or division when the numbers are decreasing, even though subtraction is clearly more natural in such a context. Some learners also check only one gap instead of verifying that the entire sequence of differences follows a uniform rule, which can lead to accepting a wrong option that fits locally but not globally. Always ensure consistency for every adjacent pair in the series.

Final Answer:

The missing term that maintains the consistent difference pattern 95, 85, 75, 65 is 205, so the correct answer is 205.