In the number series 410, 409, 413, 404, ?, find the next term by identifying the pattern of adding and subtracting consecutive square numbers.

Introduction / Context:
This series alternates up and down: 410, 409, 413, 404, ?. The changes between terms are not constant, but they suggest a link with small square numbers. Many aptitude series questions use a pattern where you add and subtract consecutive squares or other familiar values.

Given Data / Assumptions:
The known terms are:

• 410
• 409
• 413
• 404
• ?

We assume that each step involves adding or subtracting a square number such as 1^2, 2^2, 3^2, 4^2, with signs alternating between minus and plus.

Concept / Approach:
The strategy is to compute the differences between consecutive terms and check whether these differences match small perfect squares with alternating signs. Once that structure is confirmed, we can continue the pattern by using the next square number with the appropriate sign and hence find the missing term.

Step-by-Step Solution:
Step 1: Compute the difference from 410 to 409: 409 − 410 = −1, which is −1^2.Step 2: Compute the difference from 409 to 413: 413 − 409 = +4, which is +2^2.Step 3: Compute the difference from 413 to 404: 404 − 413 = −9, which is −3^2.Step 4: The pattern so far is: subtract 1^2, add 2^2, subtract 3^2. The next logical step is to add 4^2.Step 5: Compute 4^2 = 16.Step 6: Add this to the last known term: 404 + 16 = 420.

Verification / Alternative check:
We can rewrite the series according to this rule: start with 410, then subtract 1 to get 409, then add 4 to get 413, then subtract 9 to get 404, then add 16 to reach 420. This sequence uses consecutive squares 1, 4, 9 and 16, with signs alternating minus, plus, minus, plus. The pattern is perfectly consistent and leaves no room for another value in place of 420.

Why Other Options Are Wrong:
Values 412, 415 and 418 would require adding 8, 11 or 14 to 404 instead of 16. These differences are not perfect squares and do not fit into the neat sequence of 1^2, 2^2, 3^2, 4^2 with alternating signs. Therefore, choosing any of them would break the discovered rule.

Common Pitfalls:
Some students may try to find a constant difference or ratio across the series, which is not present here. Instead, recognising that small square numbers often feature in exam series is very useful. Checking whether observed differences are squares is a quick and effective technique.

Final Answer:
The next number in the series, obtained by adding 4^2 to 404, is 420.