In the following number series question, you are given the sequence 7, 26, ?, 124, 215, 342. The pattern is based on adding successively larger differences. Select the missing number from the given alternatives.

Introduction / Context:
This question tests your skill in recognising patterns in number sequences. The given series is 7, 26, ?, 124, 215, 342. One term in the middle is missing, and you must determine it using the pattern followed by the rest of the sequence. Such series often use differences that themselves follow a smaller sequence or rule, so analysing the gaps between terms is the key to solving the problem.

Given Data / Assumptions:

• Known terms: 7, 26, ?, 124, 215, 342.
• The missing term is between 26 and 124.
• Options for the missing term: 63, 65, 66 and 67.
• We assume a consistent rule for the differences between consecutive terms.
• Differences may themselves form a simple increasing pattern.

Concept / Approach:
The standard approach for such series is to compute the differences between consecutive known terms. If those differences appear irregular, we check whether the differences of the differences form a pattern. Here, when we insert a candidate for the missing term and compute all the differences, we find that the differences follow a second level increasing pattern based on adding 18, 24, 30 and 36. This reveals a systematic design behind the series rather than random jumps.

Step-by-Step Solution:
Step 1: Let the missing term be X.The sequence becomes 7, 26, X, 124, 215, 342.Step 2: Compute the known differences around X in a general way.From 7 to 26, the difference is 26 - 7 = 19.From 26 to X, the difference is X - 26.From X to 124, the difference is 124 - X.From 124 to 215, the difference is 91.From 215 to 342, the difference is 127.Step 3: Look for a pattern in differences.Suppose the differences are 19, 37, 61, 91 and 127. This suggests X - 26 = 37 and 124 - X = 61.Step 4: Solve for X.From X - 26 = 37, we get X = 26 + 37 = 63. Check 124 - 63 = 61, which is consistent.Step 5: Verify the second level pattern in differences.Differences: 19, 37, 61, 91, 127. Now the increments between these are 37 - 19 = 18, 61 - 37 = 24, 91 - 61 = 30, 127 - 91 = 36. These increments themselves increase by 6 each time, showing a clear pattern.

Verification / Alternative check:
After inserting 63 into the series, write it completely as 7, 26, 63, 124, 215, 342. Check each difference: 26 - 7 = 19, 63 - 26 = 37, 124 - 63 = 61, 215 - 124 = 91 and 342 - 215 = 127. The pattern in these differences is that each new difference is larger than the previous one by 18, 24, 30 and 36 respectively. Since this structure is smooth and consistent, 63 is the only option that fits such a neat design, confirming it as the correct missing term.

Why Other Options Are Wrong:
65, 66 and 67: If you substitute any of these instead of 63, the resulting differences do not form a consistent second level pattern with fixed increments between successive differences. This breaks the elegant structure observed with 63 and indicates that these choices are incorrect.

Common Pitfalls:
Students sometimes give up after checking only the first level differences, especially when the numbers look irregular. Another frequent mistake is trying only one candidate from the options without ensuring that a clean pattern holds for the entire sequence. For exam success, it is important to verify that your chosen term makes the sequence follow a simple and meaningful rule throughout, rather than working by guesswork.

Final Answer:
The missing number in the series is 63, so the complete sequence is 7, 26, 63, 124, 215, 342.