A numerical series is given with one term missing. Choose the correct alternative from the options that will complete the series. 7, 13, 21, 31, 43, 57, ?

Introduction / Context:
This question tests basic number series recognition, where each term follows a pattern derived from the previous term or terms. The series 7, 13, 21, 31, 43, 57 shows gradually increasing gaps between consecutive numbers. Identifying those gaps and recognising the pattern in the differences is essential for correctly predicting the next term in the sequence.

Given Data / Assumptions:

• Series provided: 7, 13, 21, 31, 43, 57, ?
• We assume the series follows a consistent rule based on arithmetic differences.
• We must determine the missing next term from the given answer options.

Concept / Approach:
A common technique for such series is to compute the differences between consecutive terms and look for a pattern in those differences. If the differences themselves form a simple progression, we can extend that progression to find the next difference, and then add it to the last known term. Here the differences increase in a consistent way, suggesting that the rule for the sequence is based on adding consecutive even numbers to each term.

Step-by-Step Solution:
Step 1: Calculate the differences between consecutive terms: 13 − 7 = 6, 21 − 13 = 8, 31 − 21 = 10, 43 − 31 = 12, 57 − 43 = 14. Step 2: Observe that the differences themselves form a clear sequence: 6, 8, 10, 12, 14. This is an arithmetic progression of even numbers increasing by 2 each time. Step 3: To continue this pattern, the next difference after 14 should be 16. Step 4: Add this next difference to the last known term: 57 + 16 = 73. Therefore, the missing term in the original series is 73.

Verification / Alternative check:
We can verify quickly by reconstructing the series using the difference pattern. Starting from 7 and adding 6, 8, 10, 12, 14, 16 sequentially, we get 7, 13, 21, 31, 43, 57, 73. This perfectly reproduces the given series and extends it consistently. Since 73 appears among the answer options and fits the established pattern of increasing even differences, it is the only logically valid continuation.

Why Other Options Are Wrong:
Option B (83) would require a difference of 26 from 57, which breaks the established pattern of increments increasing by 2 each time. Option C (78) implies a difference of 21 from 57, again incompatible with the even difference progression. Option D (63) gives a difference of only 6, which would revert back to the original smallest difference rather than continuing the two step increase. Option E (69) requires a difference of 12, repeating a previous difference instead of moving forward to 16. None of these maintain the simple pattern 6, 8, 10, 12, 14, 16 that we observed.

Common Pitfalls:
A common mistake is to guess based on approximate growth of the series without computing the exact differences. Another pitfall is to stop after noticing that the numbers seem to grow by roughly 10 on average, and then pick a random nearby value. In number series questions, always calculate the differences carefully and check whether those differences themselves follow a recognisable pattern, such as an arithmetic or geometric progression.

Final Answer:
Using the observed difference pattern of adding consecutive even numbers, the next term of the series is 73.