Let P(n) denote the statement that n2 + n is odd. Then,

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Let P(n) denote the statement that $n^{2}$ + n is odd. Then,

Options:

A . P(1) is true

B . P(2) is true

C .

P (k) \Rightarrow P (k + 1)

D .

P(n) is true \forall n ϵ N

Answer: Option C
:
C

$P (1) : 1^{2} + 1 is odd \to n o t t r u e P (2) : 2^{2} + 2 is odd \to n o t t r u e$
Suppose P(k) is true.
$k^{2} + k is odd \Rightarrow k^{2} + k = 2 m + 1 P (k + 1) : (k + 1)^{2} + (k + 1) = (k^{2} + 2 k + 1) + (k + 1) = (k^{2} + k) + (2 k + 2) = 2 m + 1 + 2 k + 2 2 (m + k + 1) + 1 \Rightarrow P(k+1) is true$

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