Prove by mathematical induction that. n^4 + 4n^2 + 11 is divisible by 16


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Question Number 78948 by TawaTawa last updated on 21/Jan/20

$Prove by mathematical induction that .$ $n^{4} + {4 n}^{2} + 11 is divisible by 16$
Commented by john santu last updated on 21/Jan/20

$l e t p (n) = n^{4} + 4 n^{2} + 11$ $f o r n = 1 \Rightarrow 1 + 4 + 11 = 16 ∣ 16 (t r u e)$ $s u p p o s e n = k s u c h t h a t$ $p (k) = k^{4} + 4 k^{2} + 11 = u (m o d 16)$ $w e c a n p r o o f t h a t f o r$ $n = k + 1 d i v i s i b l e b y 16$ $k^{4} + 4 k^{2} + 11 + {(k + 1)}^{4} + 4 (k + 1) + 11$ $n e x t . .$
Commented by TawaTawa last updated on 21/Jan/20

$It is for k + 1 i {don}^{'} t get sir$
Commented by mind is power last updated on 21/Jan/20

$e r r o r i n t h i s$ $n = 2 k {d i d n}^{'} t w o r c k e v e r$
Commented by TawaTawa last updated on 21/Jan/20

$That means the question is wrong sir ?$
Commented by mind is power last updated on 21/Jan/20

$y e a h$
Commented by john santu last updated on 21/Jan/20

$a n d . . . i n f a c t n o t p r o v e d m i s s$ $t h e e q u a t i o n e r r o r . k^{4} + 4 k^{2} + 11$ $n o t d i v i s i b l e b y 16 .$ $i . q l e t k = 2 \Rightarrow 16 + 16 + 11 \neq∣ 16$
Commented by TawaTawa last updated on 21/Jan/20

$God bless you sir$
	Answered by mind is power last updated on 21/Jan/20

	$f i r s t M e t h o d e$ $n = 2 k$ $\Rightarrow n^{4} + 4 n^{2} + 11 = 16 k^{4} + 16 k^{2} + 11 \equiv 11 (16)$ $e r r o r s i r$
	Commented by TawaTawa last updated on 21/Jan/20

	$Sir what of this$ $n^{4} + {4 n}^{2} + 11 is a multiple of 16 for all odd positive integral$
	Commented by TawaTawa last updated on 21/Jan/20

	$Help me prove this$
	Commented by mind is power last updated on 21/Jan/20

	$i f n = (2 k + 1)$ $n^{4} = {(2 k + 1)}^{4} = 16 k^{4} + 32 k^{3} + 24 k^{2} + 8 k + 1$ $n^{2} = 4 k^{2} + 4 k + 1$ $n^{4} + 4 n^{2} + 1 = 16 k^{4} + 32 k^{3} + 24 k^{2} + 8 k + 1 + 4 (4 k^{2} + 4 k + 1) + 11$ $= 16 k^{4} + 32 k^{3} + 40 k^{2} + 24 k + 16$ $= 16 (k^{4} + 2 k^{3}) + 16 + 8 k (5 k + 3) = 8 k (5 k + 3) m o d (16)$ $k (5 k + 3) = k (5 k + 5 - 2) = 5 k (k + 1) - 2 k = 2 m$ $\Rightarrow 8 k (5 k + 3) = 8.2 m = 16 m = 0 (16)$ $\Rightarrow 16 ∣ n^{4} + 4 n^{2} + 11 \Rightarrow n = (2 k + 1) i s T r u e$
	Commented by TawaTawa last updated on 22/Jan/20

	$God bless you sir$
	Commented by mind is power last updated on 22/Jan/20

	$y^{'} r e W e l c o m S i r W i t h e p l e a s u r$
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