Find the missing number in the following series: 101, 123, 147, 173, ?, 231, 263

Concept: In a number series whose first differences are not constant, check whether those differences themselves follow a simple rule (often their own arithmetic progression). Once that rule is found, the missing term equals the preceding term plus the difference the rule predicts at that position.

Application:

1. Write the given terms in order: 101, 123, 147, 173, ?, 231, 263.

2. Find the differences between consecutive known terms: 123 - 101 = 22, 147 - 123 = 24, 173 - 147 = 26.

3. These differences (22, 24, 26) increase by 2 at each step, so the differences themselves form the sequence 22, 24, 26, 28, 30, 32.

4. The difference after 173 should therefore be 26 + 2 = 28.

5. Missing term = 173 + 28 = 201.

Cross-check: Continue the same difference pattern forward from the missing term - the next difference is 28 + 2 = 30, giving 201 + 30 = 231, which matches the given term. The following difference is 30 + 2 = 32, giving 231 + 32 = 263, which also matches the given last term. Both checks confirm the missing number is 201.