In the series 31, 45, 35, 47, 39, x, 43, 51, 47, 53, find the value of x that maintains the pattern.

Introduction / Context:
This question illustrates a series where two separate number patterns are interleaved. The odd and even positions each follow their own simple rule. To find x, we must separate these two subsequences and study them individually.

Given Data / Assumptions:

• Series: 31, 45, 35, 47, 39, x, 43, 51, 47, 53.
• x appears in the 6th position.
• We assume that the sequence is organised into two interleaved arithmetic progressions.

Concept / Approach:
In interleaved series, the numbers in odd positions (1st, 3rd, 5th and so on) and even positions (2nd, 4th, 6th and so on) often form two independent arithmetic sequences. We extract these subsequences and examine their patterns separately, then recombine the results to identify the missing term.

Step-by-Step Solution:
Step 1: Write down the odd position terms: 31 (1st), 35 (3rd), 39 (5th), 43 (7th), 47 (9th).Step 2: Observe that 31, 35, 39, 43, 47 form an arithmetic sequence with a common difference of +4.Step 3: Now write down the even position terms: 45 (2nd), 47 (4th), x (6th), 51 (8th), 53 (10th).Step 4: The even subsequence starts 45, 47, ?, 51, 53. The differences 47 - 45 = 2 and 51 - 47 = 4 suggest a constant difference of +2 between every adjacent pair.Step 5: Continue this pattern: after 47, the next even term should be 47 + 2 = 49, and then 49 + 2 = 51, which matches the given term.Step 6: Therefore x must be 49.

Verification / Alternative check:
Check the two subsequences completely: odd terms 31, 35, 39, 43, 47 all differ by 4, and even terms 45, 47, 49, 51, 53 all differ by 2. This confirms that we have correctly identified the rule for both positions and that 49 is the only choice that preserves both arithmetic progressions.

Why Other Options Are Wrong:
Any other value for x such as 48, 36, 38 or 46 breaks the constant difference of 2 in the even subsequence. For example, if x were 48 then the difference from 47 would be 1, and from 48 to 51 would be 3, which is not consistent.

Common Pitfalls:
A frequent mistake is to treat the whole list as one single sequence and search for complex patterns, rather than noticing that odd and even positions behave differently. Recognising interleaving is crucial for solving many reasoning series questions efficiently.

Final Answer:
The value of x that maintains both interleaved arithmetic patterns is 49.