In the following question, a number series is given with one term missing. Select the missing number from the series that correctly continues the pattern: 1357, 3085, 5282, 8026, ?

Introduction / Context:
This is a higher level number series problem where the relation between consecutive terms is not obvious from simple differences. The large numbers 1357, 3085, 5282, 8026, ? suggest that the pattern might involve adding cubes or squares of integers. Such questions often appear in advanced reasoning sections of exams like SSC CGL and other competitive tests.

Given Data / Assumptions:

• Series: 1357, 3085, 5282, 8026, ?
• Options: 9961, 10441, 11321, 11401.
• We assume that the same rule is applied consistently between every pair of consecutive terms.

Concept / Approach:
We start by computing differences between consecutive terms. If the differences themselves match a known numerical pattern, such as cubes of consecutive integers, we can use that pattern to predict the next difference and therefore the missing term. Because the numbers are large and gaps also increase significantly, cubes are a strong candidate.

Step-by-Step Solution:
Step 1: Compute differences between consecutive terms.3085 - 1357 = 1728.5282 - 3085 = 2197.8026 - 5282 = 2744.Step 2: Recognise these differences.1728 = 12^3.2197 = 13^3.2744 = 14^3.Step 3: Extend the pattern.The cubes used are of consecutive integers 12, 13, 14, so the next should be 15^3.15^3 = 3375.Step 4: Add this cube to the last known term.Missing term = 8026 + 3375 = 11401.

Verification / Alternative check:
With the completed series 1357, 3085, 5282, 8026, 11401, the differences are 1728, 2197, 2744, 3375, which correspond to 12^3, 13^3, 14^3, 15^3. The pattern is perfectly smooth: each next term equals the previous term plus the cube of the next integer in order. This confirms 11401 as the correct continuation and rules out other options that do not match this cube based rule.

Why Other Options Are Wrong:
9961: Difference 9961 - 8026 = 1935, which is not the cube of any integer.10441: Difference 10441 - 8026 = 2415, which again is not an integer cube.11321: Difference 11321 - 8026 = 328...which does not match 15^3 and breaks the smooth cube pattern.

Common Pitfalls:
Many candidates focus only on differences without recognising them as special numbers like cubes or squares. When dealing with large integers, always check whether differences match familiar values such as 11^2, 12^2, 13^3, and so on. Another mistake is to give up too early if the transformation is not a simple arithmetic progression. Systematic checking of special forms often reveals the hidden logic.

Final Answer:
The missing term found by adding 15^3 to 8026 is 11401.