What number would replace the question mark (?) in the series below? 1.5, 4,…

Concept

In a difference-based number series, subtract each term from the next term and examine how the successive differences change.

If those differences themselves follow a constant step, extend that secondary pattern before adding the next difference to the last term.

Application

1. The successive differences are 4 − 1.5 = 2.5, 7.5 − 4 = 3.5, 12 − 7.5 = 4.5, 17.5 − 12 = 5.5, 24 − 17.5 = 6.5, and 31.5 − 24 = 7.5.

2. These differences increase by 1 each time: 2.5, 3.5, 4.5, 5.5, 6.5, 7.5.

3. Therefore, the next difference is 7.5 + 1 = 8.5.

4. Adding it to the last known term gives 31.5 + 8.5 = 40.

Cross-check

The terms can also be written as (n2 + 2n)/2 for n = 1, 2, 3, … . At n = 8, this gives (82 + 2 × 8)/2 = (64 + 16)/2 = 40, confirming the pattern.

Hence, the missing number is 40.