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

Solution

Maths sir · 3 steps

Each root satisfies the quadratic, so multiply it by α⁴ and β⁴ and add the two.

1. Root satisfies equation. Since α is a root, 5α² + 6α - 2 = 0. The same holds for β.
2. Raise the powers. Multiply the equation for α by α⁴ to get 5α⁶ + 6α⁵ - 2α⁴ = 0. Do the same for β to get 5β⁶ + 6β⁵ - 2β⁴ = 0.

   5(α⁶+β⁶) + 6(α⁵+β⁵) - 2(α⁴+β⁴) = 0

3. So. Adding gives 5S₆ + 6S₅ - 2S₄ = 0, that is 5S₆ + 6S₅ = 2S₄, which is option (a).

The idea to keep. If α, β are roots of ax²+bx+c=0, then aSₙ₊₂+bSₙ₊₁+cSₙ=0. The coefficients and signs come straight from the quadratic.

Answer from the NTA final answer key. Solution by PrepWiser.