Prove-by-mathematical-induction-that-n-4-4n-2-11-is-divisible-by-16-

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 \dots 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