In the number series 305, 338, 404, 503, 635, ( ? ), what number should come in place of the question mark to continue the pattern correctly?

Introduction / Context:
This question presents a number series and asks for the next term. Recognising the pattern behind the series is a standard requirement in many aptitude tests. The given series shows steadily increasing numbers, so examining differences between consecutive terms is a natural first step. Often, these differences themselves form a simple pattern, such as an arithmetic progression.

Given Data / Assumptions:

• The given sequence is: 305, 338, 404, 503, 635, ?.
• We are required to find the term that should replace the question mark.
• All numbers are positive integers.
• The rule governing the sequence is assumed to be consistent throughout.

Concept / Approach:
The standard method is to compute the first differences between successive terms and then look for a pattern such as a constant increment, a second-level arithmetic progression, or a multiplicative pattern. If the first differences themselves form an arithmetic progression, we can continue that progression to find the missing term. This approach keeps the reasoning systematic and avoids random guessing.

Step-by-Step Solution:
Step 1: Write the series: 305, 338, 404, 503, 635, ?.Step 2: Compute the differences between consecutive terms.Step 3: 338 - 305 = 33.Step 4: 404 - 338 = 66.Step 5: 503 - 404 = 99.Step 6: 635 - 503 = 132.Step 7: So the sequence of first differences is 33, 66, 99, 132.Step 8: Now, examine these differences. Notice that each difference increases by 33: 66 - 33 = 33, 99 - 66 = 33, 132 - 99 = 33.Step 9: This suggests the differences themselves form an arithmetic progression with first term 33 and common difference 33.Step 10: Therefore, the next difference should be 132 + 33 = 165.Step 11: The missing term is the previous term plus this new difference: ? = 635 + 165 = 800.

Verification / Alternative check:
Write out the extended series including the found term: 305, 338, 404, 503, 635, 800. The differences are now 33, 66, 99, 132, 165. The differences between these differences are 33, 33, 33, 33, showing a constant increase, which confirms that the pattern is consistent and our value 800 fits the rule correctly.

Why Other Options Are Wrong:
Option 820: Would lead to a last difference of 185, which breaks the constant increment of 33 between differences.
Option 880: Gives a last difference of 245, which does not fit the established pattern.
Option 890: Produces a difference of 255, again inconsistent with the step of adding 33 to each previous difference.
Option 760: Would yield a difference of only 125, interrupting the clear progression of 33, 66, 99, 132, 165.

Common Pitfalls:
Some test-takers try to relate terms multiplicatively or look for more complicated patterns before checking simple differences. Others may miscalculate one of the differences, causing them to miss the simple arithmetic progression pattern. It is important to carefully compute and then inspect first differences and, if needed, differences of those differences.

Final Answer:
The number that should replace the question mark in the series is 800.