If [ ] and [ ] be the coefficients of x4 and x2 respectively in the expansion of [( x + x^…

Question

If $$\alpha $$ and $$\beta $$ be the coefficients of x4 and x2
respectively in the expansion of

$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then

Accepted Answer

(x+a)n + (x -- a)n
 = 2(T1
 + T3
 + T5
 +.....)

$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$

= 2[T1
 + T3
 + T5
 + T7
]

= 2[6C0
 x6
 + 6C2

x4(x2
 -- 1) + 6C4
x2(x2
 --1)2
 + 6C6
x0(x2--1)3]

= 2[x6+ 15(x6
 -- x4) + 15x2
 (x4
 + 1 --2x2) + (x6
 -- 3x4
 +3x2
 --1)]

= 2[x6(2 + 15 + 15 + 1) + x4(--15 -- 30 --3) + x2(15 + 3)]

Coefficient of x4 = $$\alpha $$ = -96

And coefficient of x2 = $$\beta $$ = 36

$$ \therefore $$ $$\alpha  - \beta  =  - 96 - 36 =  -132$$

Solutions · 1