A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 206, 309, 412, ?, 618, 721.

Introduction / Context:
This question presents what is effectively an arithmetic series, where a constant difference is added to each term to obtain the next one. The task is to identify that difference and use it to find the missing term in the middle of the sequence. Such constant-difference patterns are fundamental in number series questions.

Given Data / Assumptions:
Given series: 206, 309, 412, ?, 618, 721.The fourth term is missing between 412 and 618.We assume that the difference between consecutive terms is constant.

Concept / Approach:
We begin by calculating the difference between the first two and next two known consecutive terms. If these differences are equal, it indicates an arithmetic progression. With the common difference established, we can compute the missing term by either adding or subtracting that difference from the nearest known term. This method is straightforward and reliable for linear patterns.

Step-by-Step Solution:
Step 1: Compute 309 - 206 = 103.Step 2: Compute 412 - 309 = 103.Step 3: The common difference between the first three terms is therefore 103, indicating an arithmetic series.Step 4: Let the missing term be x. It should be obtained by adding 103 to 412: x = 412 + 103 = 515.Step 5: Verify the later terms: 515 + 103 = 618, and 618 + 103 = 721.Step 6: Since the difference 103 holds throughout, 515 is the correct missing term.

Verification / Alternative check:
Write the full series with the candidate term: 206, 309, 412, 515, 618, 721. Now compute all differences: 309 - 206 = 103, 412 - 309 = 103, 515 - 412 = 103, 618 - 515 = 103, and 721 - 618 = 103. The constant difference of 103 remains unchanged, confirming a simple arithmetic progression and validating 515 as the correct answer.

Why Other Options Are Wrong:
Option 751 does not maintain the constant difference; 751 - 412 = 339 and 618 - 751 = -133, breaking the pattern entirely.Option 749 also yields inconsistent differences and does not allow 618 and 721 to be produced by repeatedly adding the same number.Option 514 would give 514 - 412 = 102, and 618 - 514 = 104, so the difference is not constant and fails to form a proper arithmetic series.

Common Pitfalls:
One common error is miscalculating the difference, especially with three-digit numbers. Another is assuming a more complex pattern than necessary when a simple arithmetic series with a constant difference is sufficient. Always verify differences between multiple consecutive terms before deciding on the pattern and computing the missing value.

Final Answer:
The missing number that completes the arithmetic series is 515.