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Question

If α,β are the roots of quadratic equation 6x22x+1=0 and Sn=αn+βn, then which of the following statement(s) is/are true ?

A
6Sn+12Sn+Sn1=0
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B
6Sn+12Sn+Sn3=0
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C
2S66S7S5=1
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D
2S66S7S5=3
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Solution

The correct option is C 2S66S7S5=1
Given equation 6x22x+1=0 has roots α,β
6α22α+1=0(6α2+1α)=0αn(6α2+1α)=06αn+12αn+αn1=0 (1)

Similarly, with root β
6βn+12βn+βn1=0 (2)

Adding equation(1) and (2),
6(αn+1+βn+1)2(αn+βn)+(αn1+βn1)=06Sn+12Sn+Sn1=0 (Sn=αn+βn)

Now, Sn1=2Sn6Sn+1
2Sn6Sn+1Sn1=1 (3)
Putting the value of n=6 in the equation (3), we get
2S66S7S5=1

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