The next number in the series 2, 5, 9, 19, 37, ? will be

Concept

A number series is produced by one fixed rule that builds every term out of the terms before it. A very common family is the linear recurrence: each term is a fixed multiple of the previous term plus a small constant, and that constant is allowed to alternate in sign from one step to the next. The way to identify such a rule is to first compare the size of consecutive terms to find the multiplier, and then read off the small residual that the multiplier leaves over.

Applying the concept to this series

1. Compare consecutive terms. 5 ÷ 2 = 2.5, 9 ÷ 5 = 1.8, 19 ÷ 9 ≈ 2.1 and 37 ÷ 19 ≈ 1.95 all sit close to 2, so test the rule an = 2 × an−1 + c, where c is a small constant.

2. First step: 2 × 2 = 4, while the series gives 5. The residual is +1.

3. Second step: 2 × 5 = 10, while the series gives 9. The residual is −1.

4. Third step: 2 × 9 = 18, while the series gives 19. The residual is +1.

5. Fourth step: 2 × 19 = 38, while the series gives 37. The residual is −1.

6. The residuals run +1, −1, +1, −1, so the alternation makes the next residual +1. Therefore the next term is 2 × 37 + 1 = 75.

Cross-check with an independent rule

A completely different recurrence reproduces the same series: each term equals the previous term plus twice the term before that, i.e. an = an−1 + 2 × an−2. Checking it on every given term:

• 5 + 2 × 2 = 9

• 9 + 2 × 5 = 19

• 19 + 2 × 9 = 37

• 37 + 2 × 19 = 75

Two independent rules that both reproduce every given term also agree on the continuation, so the series goes on with 75. Since 75 itself appears among the listed values, the “none of the above” choice does not apply here.