If $n$ is a positive integer then statement $P (n) : n^{2} + n^{3} - 2 n$ is divisible $24$ by is true for [ $k ϵ N$ ]

P (4 k)

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P (2 k)

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P (4 k + 1)

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N o n e o f t h e s e

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Solution

The correct option is D $N o n e o f t h e s e$
$\begin{matrix} P (n) = n^{2} + n^{3} - 2 n i s d i v i s i b l e b y 24 c h e c k a t n = 1 P (1) = 0 I t i s t r u e P (4 k) = {(4 k)}^{2} + {(4 k)}^{3} - 2 \times 4 k P (4 k) = 16 k^{2} + 64 k^{3} - 8 k P (4 k) = 8 k (2 k + 8 k^{2} - 1) P (4 k) = 8 k (8 k^{2} + 4 k - 2 k - 1) P (4 k) = 8 k (4 k - 1) (2 k + 1) i f K \in N P u t K = 1, 2, 3 P (4) = 8 \times 3 \times 3 = 72 i t i s d i v i s i b l e b y x_{1} K = 2 \end{matrix}$
$P (8) = 16 \times 7 \times 5 =$ It is does not divisible by 24
$P (2 k), P (4 k, 1)$ are also does not divisible by 24.

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