If α, β are the roots of the equation, x²-x-1=0 and Sₙ=2023 αⁿ+2024 βⁿ, then :

Solution

Maths sir · 3 steps

Each root satisfies x² = x + 1; multiply by x¹⁰ and the whole sequence inherits that rule.

1. Use the equation. Since α is a root, α² = α + 1. Multiply both sides by α¹⁰ to get α¹² = α¹¹ + α¹⁰. The same holds for β: β¹² = β¹¹ + β¹⁰.
2. Attach the weights. Multiply the α relation by 2023 and the β relation by 2024, then add them. The constants ride along unchanged because each term on both sides carries the same weight.

   2023α¹² + 2024β¹² = (2023α¹¹ + 2024β¹¹) + (2023α¹⁰ + 2024β¹⁰)

3. So. This reads S₁₂ = S₁₁ + S₁₀, which is option (b).

The idea to keep. If α, β are roots of x² = px + q, then any Sₙ = aαⁿ + bβⁿ obeys Sₙ₊₂ = pSₙ₊₁ + qSₙ, whatever the constants a and b are.

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