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.

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