Use the Principle of Mathematical Induction to prove that $$2 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 0$$

Name: Use the Principle of Mathematical Induction to prove that \[2 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 0\]
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Question

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The Answer is

P ( 0 ) : 2 \mid 0 ^ { 2 } + 0

, which is true since

2 \mid 0 . \quad P ( k ) \rightarrow P ( k + 1 ) : ( k + 1 ) ^ { 2 } + ( k + 1 ) = \left( k ^ { 2 } + k \right) + 2 ( k + 1 )

, which is divisible by 2 since

2 \mid k ^ { 2 } + k

and

2 \mid 2 ( k + 1 )

.

Use the Principle of Mathematical Induction to Prove That 2∣(n2+n) for all n≥02 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 02∣(n2+n) for all n≥0

Use the Principle of Mathematical Induction to Prove That $2 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 0$