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Question
If P (n) is the statement "n2 + n is even", and if P (r) is true, then P (r + 1) is true.
Solution
\[P(n): n^2 + n is even . \]
\[Also, \]
\[P(r) is true . \]
\[Thus, r^2 + r is even . \]
\[To prove: P(r + 1) is true . \]
\[Now, \]
\[P(r + 1) = (r + 1 )^2 + r + 1\]
\[ = r^2 + 1 + 2r + r + 1 \]
\[ = r^2 + 3r + 2\]
\[ = r^2 + r + 2r + 2\]
\[ = P(r) + 2(r + 1)\]
\[P(r) \text{ is even } . \]
\[Also, 2(r + 1)\text{ is even, as it is a multiple of 2 } . \]
\[\text{ Therefore, P(r + 1) is even and true } . \]
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