If P (n) is the statement
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,

$P(r+1)=(r+1)^{2}+r+1$

$=r^{2}+1+2 r+r+1$

$=r^{2}+3 r+2$

$=r^{2}+r+2 r+2$

$=P(r)+2(r+1)$

$P(r)$ is even.

Also, $2(r+1)$ is even, $a s$ it is a multiple of 2 .

Therefore, $P(r+1)$ is even and true.

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