Expansions-9

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If a 2 + 1 / a 2 = 23 , find the value of a 3 + 1 / a 3

Solution :

a 2 + 1 / a 2 = 23

=> a 2 + 1 / a 2 + 2 = 23 + 2 = 25

=> ( a + 1 / a ) 2 = 25

=> ( a + 1 / a ) = ± 5

When a + 1 / a = 5 :

a 3 + 1 / a 3 = ( a + 1 / a ) 3 - 3 ( a + 1 / a )

= ( 5 ) 3 - 3 X 5

= 110

When a + 1 / a = - 5 :

a 3 + 1 / a 3 = ( a + 1 / a ) 3 - 3 ( a + 1 / a )

= ( - 5 ) 3 - 3 X ( - 5 ) = - 125 + 15

= - 110

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