In the number series 514, 470, ?, 415, 404, find the missing term by identifying the regular pattern in the decreasing differences.

Introduction / Context:
This problem presents a descending number series with one missing term: 514, 470, ?, 415, 404. The numbers are fairly close together, which suggests that the differences between consecutive terms follow a neat, stepwise pattern. Our job is to uncover this pattern and use it to find the unknown term.

Given Data / Assumptions:
The series, including the unknown, is:

• 514
• 470
• ?
• 415
• 404

We assume that there is a consistent rule for the amount subtracted at each step, likely involving simple multiples of a small integer.

Concept / Approach:
A natural strategy is to look at the differences between known consecutive terms and see whether these differences follow a pattern such as a decreasing arithmetic sequence. If we can express each step as subtracting a multiple of some base number, the missing term can be found by continuing that pattern.

Step-by-Step Solution:
Step 1: Compute the difference from the first term to the second term.514 − 470 = 44.Step 2: Compute the difference from the fourth term to the fifth term.415 − 404 = 11.Step 3: Observe that 44 and 11 look like 4 * 11 and 1 * 11 respectively. This hints at subtracting multiples of 11 with coefficients 4, 3, 2 and 1 in order.Step 4: Following this idea, the sequence of subtractions could be: −44, −33, −22, −11.Step 5: Apply this pattern step by step.From 514 subtract 44 to get 514 − 44 = 470.From 470 subtract 33 to get 470 − 33 = 437.From 437 subtract 22 to get 437 − 22 = 415.From 415 subtract 11 to get 415 − 11 = 404.

Verification / Alternative check:
This reconstruction gives the complete series 514, 470, 437, 415, 404 with a clear rule: each term is obtained by subtracting multiples of 11 that decrease by 11 each time (44, 33, 22, 11). Every step fits perfectly, which confirms that the missing term must be 437 in order to preserve this regular pattern.

Why Other Options Are Wrong:
Values 441, 426 and 420 do not fit into the coherent subtraction pattern of −44, −33, −22, −11. If any of them were used in place of 437, the required differences to reach 415 and 404 would be irregular and would not be simple multiples of 11 in decreasing order. Hence they are inconsistent with the logic of the series.

Common Pitfalls:
Students may try to guess the missing number by rough estimation instead of systematically checking the differences. For series with regular step decreases, always test whether the differences can be expressed as multiples of a common number. That approach quickly reveals the underlying rule and leads to a precise answer.

Final Answer:
The missing number in the series is 437.