The next term in the series is : 2, 5, 9, 19, 37, ?

Concept: in a number series each term is generated from the one before it by a fixed rule. When the successive differences show no simple pattern, test a multiplicative recurrence — a term of the form 2 x (previous term) ± c — and check whether the sign of the constant c stays fixed or alternates from step to step.

Application to the given series:

1. 2 x 2 + 1 = 5

2. 2 x 5 - 1 = 9

3. 2 x 9 + 1 = 19

4. 2 x 19 - 1 = 37

5. The constants run +1, -1, +1, -1, so the next step takes +1: 2 x 37 + 1 = 75.

Cross-check: the successive differences are 3, 4, 10, 18, which do not yield an equally simple continuation; the alternating doubling recurrence is the intended compact rule. Running the multiplicative rule backwards confirms it — (75 - 1) / 2 = 37 and (37 + 1) / 2 = 19 — so the alternating doubling rule reproduces the series in both directions.

Contrast with the nearby values: 2 x 37 - 1 = 73 repeats the -1 constant used at the previous step instead of alternating back to +1, while 78 and 80 are 4 and 6 more than 2 x 37, amounts no step of this series produces.

The next term is 75.