if a=3+√5/2 then find the value of a^2+1/a^2
Answers
Answer:
7
Step-by-step explanation:
Given ----> a = ( 3 + √5 ) / 2
To find -----> Value of a² + ( 1 / a² )
Solution ----> ATQ,
a = ( 3 + √5 ) / 2
1 / a = 1 / ( 3 + √5 ) / 2
= 2 / ( 3 + √5 )
= 2 ( 3 - √5 ) / ( 3 + √5 ) ( 3 - √5 )
= 2 ( 3 - √5 ) / { ( 3 )² - ( √5 )² }
= 2 ( 3 - √5 ) / ( 9 - 5 )
= 2 ( 3 - √5 ) / 4
1 / a = ( 3 - √5 ) / 2
Now,
a + ( 1 / a ) = ( 3 + √5 ) / 2 + ( 3 - √5 ) / 2
= ( 3 + √5 + 3 - √5 ) / 2
= 6 / 2
=>a + ( 1 / a ) = 3
Squaring both sides we get,
=> { a + ( 1 / a ) }² = ( 3 )²
We have an identity ,
( a + b )² = a² + b² + 2ab , applying it here we get,
=> a² + ( 1 / a )² + 2 a ( 1 / a ) = 9
=> a² + ( 1 /a² ) + 2 = 9
=> a² + ( 1 / a² ) = 9 - 2
=> a² + ( 1 / a² ) = 7