Find wrong number in series: 1, 8, 64, 125, 216

Concept: The series consists of perfect cubes of natural numbers — n3. In an unbroken cube series, the cube root increases by exactly 1 from each term to the next — every consecutive root-to-root difference must equal 1.

Application: Find the cube root of each term and its difference from the previous root:

1. 1 = 13 — root 1 (starting point).

2. 8 = 23 — root 2; difference from the previous root = 2 − 1 = 1 (correct).

3. 64 = 43 — root 4; difference from the previous root = 4 − 2 = 2 (should be 1).

4. 125 = 53 — root 5; difference from the previous root = 5 − 4 = 1 (correct).

5. 216 = 63 — root 6; difference from the previous root = 6 − 5 = 1 (correct).

Every consecutive difference equals 1 except the one entering the third term, which jumps by 2 instead. That single deviation pins down the third term, 64, as the number that breaks the pattern; it should be 33 = 27 to keep every difference equal to 1.

Cross-check: The root-to-root differences across the series are 1, 2, 1, 1 — exactly one of the four differences (the second one, entering the third term) is not equal to 1, and it is the only irregularity in the entire series. This isolates 64 as the unique term responsible for the deviation, confirming it is the wrong number.