Answer:
see explanation
Step-by-step explanation:
A difference of cubes factors in general as
a³ - b³ = (a - b)(a² + ab + b²)
Thus
(n + 1)³ - n³ → with a = n + 1 and b = n
= (n + 1 - n)((n + 1)² + n(n + 1) + n²) ← simplifying
= 1 (n² + 2n + 1 + n² + n + n²)
= 3n² + 3n + 1 ← factor out 3n from the first 2 terms
= 3n(n + 1) + 1
= 6m + 1 ( since n or n + 1 is even )
= odd
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6m is always even for m ∈ Z ( the set of integers )
Thus 6m + 1 is always odd
Example
m = 1 → 6 + 1 = 7
m = 2 → 12 + 1 = 13
m = 3 → 18 + 1 = 19
