Question

Let α and β be the roots of the equation, 5x² + 6x – 2 = 0. If S_n = αⁿ + βⁿ, n = 1, 2, 3, ....., then ______.

Options

• 5S₆ + 6S₅ + 2S₄ = 0

• 6S₆ + 5S₅ = 2S₄

• 6S₆ + 5S₅ + 2S₄ = 0

• 5S₆ + 6S₅ = 2S₄

Answer 1

Let α and β be the roots of the equation, 5x² + 6x – 2 = 0. If S_n = αⁿ + βⁿ, n = 1, 2, 3, ....., then `underlinebb(5S_6 + 6S_5 = 2S_4)`.

Explanation:

5x² + 6x – 2 = 0

α + β = `(-6)/5`

αβ = `(-2)/5`

and 5x² + 6x – 2 = 0

⇒ 5α² = –6α + 2

Similarly, 5β² = –6β + 2

= αⁿ + βⁿ

S₆ = α⁶ + β⁶

S₅ = α⁵ + β⁵

S₄ = α⁴ + β⁴

6S₅ – 2S₄ = 6(α⁵ + β⁵) – 2(α⁴ + β⁴)

= α⁴(6α – 2) + β⁴(6β – 2)

= α⁴(–5α²) + β⁴(–5β²)

= –5(α⁶ + β⁶)

= –5S₆

⇒ 5S₆ + 6S₅ = 2S₄