Question

Can you please help me with this..

Answer 1

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