What is the next number in the following sequence: 3/7, 5/21, 7/43, 9/73,…

Concept: When a sequence of fractions does not have an obvious single-step pattern, examine the numerator and denominator series separately. If the denominator's consecutive differences are not constant, compute the differences of those differences (the second differences) — a constant second difference means the denominator follows a quadratic growth, and the next term is found by extending that constant second difference one step further.

1. The numerators 3, 5, 7, 9, 11 are consecutive odd numbers, each 2 more than the one before, so the next numerator is 11 + 2 = 13.

2. The denominators are 7, 21, 43, 73, 111. Their consecutive differences are 21 − 7 = 14, 43 − 21 = 22, 73 − 43 = 30, and 111 − 73 = 38.

3. These differences themselves increase by a constant 8 each time (22 − 14 = 8, 30 − 22 = 8, 38 − 30 = 8), so the next difference is 38 + 8 = 46.

4. Adding this to the last denominator gives the next denominator: 111 + 46 = 157.

Cross-check: the differences 14, 22, 30, 38, 46 themselves form an arithmetic progression with first term 14 and common difference 8, so the fifth difference is 14 + 8 × 4 = 46 — the same value obtained above, confirming the next denominator is 157.

So the next term of the sequence is 13/157.