if x^2+x-1=0 find x^5+1/x^5
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Step-by-step explanation:
Let roots of given equation be α and β
α.β=1
⇒β=
α
1
Hence α
5
+(
α
1
)
5
is same as α
5
+β
5
Now
α
2
−5α+1=0
β
2
−5β+1=0
multiplying by α
n−2
and β
n−2
and reaarranging
α
n
=5α
n−1
−α
n−2
β
n
=5β
n−1
−β
n−2
Adding we get
α
n
+β
n
=5α
n−1
+5β
n−1
−α
n−2
−β
n−2
let α
n
+β
n
be A
n
Then
A
n
=5A
n−1
−A
n−2
A
0
=2 and
A
1
=5 sum of roots.
Hence
A
2
=25−2=23
A
3
=115−5=110
A
4
=550−23=527
A
5
=2635−110=2525
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