Question

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★ STANDARD QUESTION 12 ★

The answer with highest efficiency will get marked THE Brainliest , And will be offered an special invitation card

Question is -

Let α and β be the roots of x² - 6x - 2 = 0 , with α > β , If aₙ = αⁿ- βⁿ for n ≥ 1 , then find the value of
a₁₀ - 2a₈ : 2a₉


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Answer 1

◆ Given equation :-

=) x² - 6x - 2 = 0

and, aₙ = αⁿ- βⁿ

so α and β are roots of given eqn, so it satisfy the equation,

now
$\alpha { }^{2} - 6 \alpha - 2 = 0$

so α^2 - 2 = 6α ----------(1) similarly for (β)

=) a₁₀ - 2a₈ : 2a₉ =
$\alpha {}^{10} - \beta {}^{10} - 2( \alpha {}^{8} - \beta {}^{8} ) \ \\ \div 2( \alpha {}^{9} - \beta {}^{9} )$
==) {α^8(α^2 -2) - β^8(β^2 -2)} /2(α^9 - β^9)

=) (α^8 × 6α - β^8×6β)/2(α^9 - β^9)

=) (6α^9 - 6β^9)/2(α^9 - β^9)

=) 6(α^9 - β^9)/2(α^9 - β^9) = 3

=) a₁₀ - 2a₈ : 2a₉ = 3 ans
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#shreya