Prove by induction n^3+(n+1)^3+(n+2)^3 is divisible by 9

Given

n³ + (n + 1)³ + (n + 2)³ is divisible by 9 ∀ n ∈ N

Proof

We will prove the given statement by induction

STEP 1

n = 1

1³ + (1 + 1)³ + (1 + 2)³= 1³ + 2³ + 3³

= 1 + 8 + 27

= 36

36 is divisible by 9. Therefore, the statement is true for n = 1

STEP 2

Let the given statement be true for n = k

k³ + (k + 1)³ + (k+ 2)³ = 9x

Now, we need to prove that if the statement is true for n = k then it is also true for n = k + 1

(k + 1)³ + (k+ 2)³ + (k+ 3)³ = ( (k + 1)³ + (k+ 2)³ ) + (k+ 3)³

= 9x – k³ + (k+ 3)³

= 9x – k³ + k³ + 3³ + 3*k²*3 + 3*k*3²

= 9x + 27 + 9k² + 27k

= 9( x + 3 + k² + 3k )

9( x + 3 + k² + 3k ) is divisible by 9.

Therefore, we can say that if the given statement is true for n = k, then it is also true for n = k + 1. Hence, by the principle of mathematical induction, the given statement is true ∀ n ∈ N.