Prove by exhaustion (cases)

1. A student was asked to prove by exhaustion that
   if \(n\) is an integer then \(2 n ^ { 2 } + n + 1\) is not divisible by 3

The start of the student's proof is shown in the box below. Consider the case when \(n = 3 k\) $$2 n ^ { 2 } + n + 1 = 18 k ^ { 2 } + 3 k + 1 = 3 \left( 6 k ^ { 2 } + k \right) + 1$$ which is not divisible by 3 Complete this proof.

7 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .

7

1. Explain why \(n ( n + 1 )\) is a multiple of 2 when \(n\) is an integer.
2. 1. Given that $$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$ show that \(\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\), where \(k\) is a positive integer, is a multiple of 6 .
   2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 6 for all integers \(n \geqslant 1\).

5 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .