Prove by induction that the sum of the cubes of three consecutive natural numbers is divisible by `9.`

Explanation

Let P(n) be the statement given by

P(n): Sum of the cubes of three consecutive natural numbers starting from n is divisible by 9.


Step I: P(1): Sum of the cubes of first three consecutive natural numbers is divisible by 9.

since `1^3+2^3+3^3=36,` which is divisible by 9.

∴P(1) is true.


Step II: Let P(m) be true. Then, sum of the cubes of three consecutive natural numbers starting with m is divisible by 9.

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