Let
$\alpha$ and
$\beta$ be the roots of the equation
5x2 + 6x – 2 = 0. If Sn
=
$\alpha$n +
$\beta$n, n = 1, 2, 3....,
then :
Solution
$\alpha$ and
$\beta$ be the roots of the equation
<br>5x<sup>2</sup> + 6x – 2 = 0.
<br><br>$\Rightarrow$ 5$\alpha$<sup>2</sup> + 6$\alpha$ - 2 = 0
<br><br>$\Rightarrow$ 5$\alpha$<sup>n + 2</sup> + 6$\alpha$<sup>n + 2</sup> - 2$\alpha$<sup>n</sup> = 0 ......(1)
<br><br>(By multiplying $\alpha$<sup>n</sup>)
<br><br>Similarly 5$\beta$<sup>n + 2</sup> + 6$\beta$<sup>n + 2</sup> - 2$\beta$<sup>n</sup> = 0 ......(2)
<br><br>By adding (1) & (2)
<br><br>5S<sub>n+2</sub> + 6S<sub>n+1</sub> – 2S<sub>n</sub> = 0
<br><br>For n = 4
<br><br>5S<sub>6</sub>
+ 6S<sub>5</sub>
= 2S<sub>4</sub>
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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