What number should replace '?' in the series: 397, 398, 371, 496, ?, 882?

Concept

In a number series, compare consecutive terms by calculating their differences.

When the signs alternate and the magnitudes follow consecutive odd cubes, extend the same pattern to the missing position.

Application

1. The first differences are 398 − 397 = 1 = 13, 371 − 398 = −27 = −33, and 496 − 371 = 125 = 53.

2. Thus the operations are +13, −33, +53, −73, +93.

3. The missing term is 496 − 73 = 496 − 343 = 153.

4. The next term is 153 + 93 = 153 + 729 = 882.

Cross-check

The calculated value reproduces the given final term, so the missing number is 153.