If are the roots of . If for . Then the value of is

Question

If $α, β$ are the roots of $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ . Then the value of $\frac{a_{12} - 2 a_{10}}{3 a_{11}}$ is _________

Smart Achievers · Accepted Answer

(1)

$α^{12} - 6 α^{11} - 2 α^{10} = 0,$ $α^{12} - 2 α^{10} = 6 α^{11},$ $β^{12} - 2 β^{10} = 6 β^{11}$

Solution

Q.
If $α, β$ are the roots of $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ . Then the value of $\frac{a_{12} - 2 a_{10}}{3 a_{11}}$ is _________

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Ans.
(1)

$α^{12} - 6 α^{11} - 2 α^{10} = 0,$ $α^{12} - 2 α^{10} = 6 α^{11},$ $β^{12} - 2 β^{10} = 6 β^{11}$

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Solution

Q. If α,β are the roots of x2-6x-2=0. If an=αn-βn for n≥1. Then the value of a12-2a103a11 is _________

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3 4

4 3

Ans. (1) α12-6α11-2α10=0, α12-2α10=6α11, β12-2β10=6β11

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Q.
If $α, β$ are the roots of $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ . Then the value of $\frac{a_{12} - 2 a_{10}}{3 a_{11}}$ is _________

Ans.
(1)

$α^{12} - 6 α^{11} - 2 α^{10} = 0,$ $α^{12} - 2 α^{10} = 6 α^{11},$ $β^{12} - 2 β^{10} = 6 β^{11}$