Let α and β be the roots of equation x 2 - 6 x - 2 = 0 . If a n = α n - β n ,   ∀   n ≥ 1 , then the value of a 1 0 - 2 a 8 2 a 9 is equal to

Question

Let &#945; and &#946; be the roots of equation x 2 - 6 x - 2 = 0 . If a n = &#945; n - &#946; n , &#160; &#8704; &#160; n &#8805; 1 , then the value of a 1 0 - 2 a 8 2 a 9 is equal to

Aicharya · Accepted Answer

Given, &#945; , &#160; &#946; are roots of x 2 - 6 x - 2 = 0 . &#8658;&#160; &#945; 2 - 6 &#945; - 2 = 0 and &#946; 2 - 6 &#946; - 2 = 0 &#8658;&#160;&#160; &#945; 2 - 2 = 6 &#945; and &#946; 2 - 2 = 6 &#946; &#160; &#160; . . . 1 a 1 0 - 2 a 8 2 a 9 = &#945; 10 - &#946; 10 - 2 &#945; 8 - &#946; 8 2 &#945; 9 - &#946; 9 = &#945; 1 0 - 2 &#945; 8 - &#946; 1 0 - 2 &#946; 8 2 &#945; 9 - &#946; 9 = &#945; 8 &#945; 2 - 2 - &#946; 8 &#946; 2 - 2 2 &#945; 9 - &#946; 9 = &#945; 8 6 &#945; - &#946; 8 6 &#946; 2 &#945; 9 - &#946; 9 [ &#8757; From (1)] = 6 &#945; 9 - 6 &#946; 9 2 &#945; 9 - &#946; 9 = 3

Let α and β be the roots of equation x 2 - 6 x - 2 = 0 . If a n = α n - β n ,  …

Solution

Let &#945; and &#946; be the roots of equation x 2 - 6 x - 2 = 0 . If a n = &#945; n - &#946; n , &#160;…

Solution

Let α and β be the roots of equation x 2 - 6 x - 2 = 0 . If a n = α n - β n , …