- 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.