Question

if a =( 3+√5)÷2 then find the value of a²+1/a²

Answer 1

Given,

a = ( 3 + √5 ) /2 

So, a² = { ( 3 + √5 ) / 2 ) }²

             9 + 5 + 6√5
          = -------------------------- 
                        4

            14 + 6√5
         = ----------------
                   4

           2 ( 7 + 3√5 )
        = ---------------------
                   4

       = ( 7 + 3√5 ) / 2.

Now , 1 / a² = 2 / ( 7 + 3√5 )

By multiplying numerator and denominator by ( 7 - 3√5 ).

 1 / a² = 2 ( 7 - 3√5 ) / ( 7 + 3√5 ) ( 7 - 3√5 ).

          = 2 ( 7 - 3√5 ) / { 7² - ( 3√5 )² }

          = 2 ( 7 - 3√5 ) / ( 49 - 45 )

          = 2 ( 7 - 3√5 ) / 4

          = ( 7 - 3√5 ) / 2.


Now ,

 = a² + 1/a²

 = { ( 7 + 3√5 ) / 2 } + { ( 7 - 3√5 ) / 2 }


    7 + 3√5  + 7 - 3√5
= -----------------------------
                2

           14
=  ------------------
             2

= 7.

Answer 2

Answer:

Given

a= (3+√5) /2

so , a²={(3+√5)/2}²

=9+5+6√5

_________

4

= 14+6√5

________

4

= 2(7+3√5)

________

4

= (7+3√5) /2

By multiplying numerator and denominator by (7-3√5).

1/a²= 2(7-3√5)/( 7+3√5) (7-3√5)

=2(7-3√5)/{7²-(3√5)²}

=2(7-3√5)/(49-45)

=2(7-3√5)/4

=(7-3√5)/2

Now

= a²+/a²

={(7+3√5) /2} +{7-3√5) /2}

= 7+3√5+7-3√5

______________

2

= 14

___

2

=7