Number series – find the missing term: 156, 506, ( ? ), 1806. Identify the value that correctly completes the pattern and replaces the question mark.

Introduction / Context:
Number-series questions often rely on consistent growth rules such as fixed differences, second-order differences, or alternating operations. The task is to infer the governing pattern and apply it to find the missing term between 506 and 1806.

Given Data / Assumptions:

• Series: 156, 506, ?, 1806.
• Exactly one standard monotone pattern is intended (no alternating sub-series are indicated).
• We must choose from the provided options.

Concept / Approach:
The cleanest hypothesis is an arithmetic pattern with increasing differences. Compute first differences and check if they themselves grow by a constant amount. This “second-order constant difference” model is common in test-series design and leads to linear growth in the increments across consecutive terms.

Step-by-Step Solution:
Let the missing term be x. Then differences are: 506 − 156 = 350, x − 506, and 1806 − x.Assume a constant step-up of +200 in the differences: 350, 550, 750.Solve x − 506 = 550 → x = 1056.Check final difference: 1806 − 1056 = 750, which matches the expected +200 step-up from 550.

Verification / Alternative check:
Differences become 350 → 550 → 750, which rise by a constant 200. This second-order consistency verifies x = 1056 and rules out other options (856, 1456, 1506), which would break the constant 200 increase.

Why Other Options Are Wrong:
856: Would give differences 350, 350, 950 (no constant step in the increments).
1456: Would give 350, 950, 350 (pattern breaks and is non-monotone).
1506: Would give 350, 1000, 300 (again inconsistent growth of differences).

Common Pitfalls:
Jumping to a multiplicative pattern due to large numbers; however, steady second-order differences are simpler and fit all four terms exactly.

Final Answer:
1056