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JEE Main202427 Jan 2024Evening ShiftMathematicsQuadratic EquationActual

If α , β are the roots of the equation, x 2 - x - 1 = 0 and S n = 2023 α n + 2024 β n , then

Options

  1. A2 S 12 = S 11 + S 10
  2. BS 12 = S 11 + S 10
  3. C2 S 11 = S 12 + S 10
  4. DS 11 = S 10 + S 12

Correct answer

B. S 12 = S 11 + S 10

Step-by-step solution

Given: α , β are roots of x 2 - x - 1 = 0 . ⇒ α 2 - α - 1 = 0 ⇒ α 2 = α + 1 . . . i Also, S n = 2023 α n + 2024 β n ⇒ S n - 1 + S n - 2 = 2023 α n - 1 + 2024 β n - 1 + 2023 α n - 2 + 2024 β n - 2 ⇒ S n - 1 + S n - 2 = 2023 α n - 2 1 + α + 2024 β n - 2 1 + β Using equation i , ⇒ S n - 1 + S n - 2 = 2023 α n - 2 α 2 + 2024 β n - 2 β 2 ⇒ S n - 1 + S n - 2 = 2023 α n + 2024 β n ⇒ S n - 1 + S n - 2 = S n Putting, n = 12 ⇒ S 12 = S 11 + S 10

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