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Question
If `a = (3 + sqrt(5))/2`, then find the value of `a^2 + 1/a^2`.
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Solution
Given: `a = (3 + sqrt(5))/2`
The value of a2 will be `a^2 = ((3 + sqrt(5))/2)^2`
= `(9 + 5 + 6sqrt(5))/4`
= `(14 + 6sqrt(5))/4`
= `(7 + 3sqrt(5))/2`
Now, `1/a^2 = 2/(7 + 3sqrt(5))`
= `2/(7 + 3sqrt(5)) xx (7 - 3sqrt(5))/(7 - 3sqrt(5))`
= `(2(7 - 3sqrt(5)))/(7^2 - (3sqrt(5))^2`
= `(2(7 - 3sqrt(5)))/(49 - 45)`
= `(2(7 - 3sqrt(5)))/4`
= `(7 - 3sqrt(5))/2`
The value of `a^2 + 1/a^2` is
`a^2 + 1/a^2 = (7 + 3sqrt(5))/2 + (7 - 3sqrt(5))/2`
= `(7 + 3sqrt(5) + 7 - 3sqrt(5))/2`
= `14/2`
= 7
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