Without-using-tables-find-tha-value-of-5-2-6-5-2-6-8-5-

Question Number 51235 by peter frank last updated on 25/Dec/18

W i t h o u t u s i n g t a b l e s,

f i n d t h a v a l u e o f

\frac{{(\sqrt{5} + 2)}^{6} - {(\sqrt{5} - 2)}^{6}}{8 \sqrt{5}}

Answered by mr W last updated on 25/Dec/18

l e t a = \sqrt{5} + 2

\sqrt{5} - 2 = \frac{1}{\sqrt{5} + 2} = \frac{1}{a}

a^{6} - \frac{1}{a^{6}} = (a^{3} + \frac{1}{a^{3}}) (a^{3} - \frac{1}{a^{3}}) = (a + \frac{1}{a}) (a^{2} + \frac{1}{a^{2}} - 1) (a - \frac{1}{a}) (a^{2} + \frac{1}{a^{2}} + 1)

= (a^{2} - \frac{1}{a^{2}}) [{(a^{2} + \frac{1}{a^{2}})}^{2} - 1]

= (8 \sqrt{5}) (18^{2} - 1)

\Rightarrow \frac{{(\sqrt{5} + 2)}^{6} - {(\sqrt{5} - 2)}^{6}}{8 \sqrt{5}} = 18^{2} - 1

= 17 \times 19 = 323

Commented by peter frank last updated on 25/Dec/18

m u c h r e s p e c t s i r n i c e w o r k

Commented by malwaan last updated on 25/Dec/18

great work

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Dec/18

{(a + b)}^{6} = a^{6} + 6 c_{1} a^{5} b + 6 c_{2} a^{4} b^{2} + 6 c_{3} a^{3} b^{3} + 6 c_{4} a^{2} b^{4} + 6 c_{5} {a b}^{5} + b^{6}

{(a - b)}^{6} = a^{6} - 6 c_{1} a^{5} b + 6 c_{2} a^{4} b^{2} - 6 c_{3} a^{3} b^{3} + 6 c_{4} a^{2} b^{4} - 6 c_{5} {a b}^{5} + b^{6}

{(a + b)}^{6} - {(a - b)}^{6}

= 2 [6 c_{1} a^{5} b + 6 c_{3} a^{3} b^{3} + 6 c_{5} {a b}^{5}]

= 2 [6 a^{5} b + 20 a^{3} b^{3} + 6 {a b}^{5}]

= 12 {(\sqrt{5})}^{5} (2) + 40 {(\sqrt{5})}^{3} {(2)}^{3} + 12 (\sqrt{5}) {(2)}^{5}

= 12 \times 2 \times 25 \times \sqrt{5} + 40 \times 8 \times 5 \sqrt{5} + 32 \times 12 \times \sqrt{5}

= 8 \sqrt{5} (75 + 200 + 48)

a n s i s 75 + 200 + 48 = 323

Commented by peter frank last updated on 25/Dec/18

p e r f e c t s i r . b i n o m i a l e x p a n s i o n

i h e s i t a t e t o a p p l y t h a t

m e t h o d