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JEE MainMathematicsQuadratic Equation

A sequence is defined as A_n = 2024 p^n + 2025 q^n , where p and q are distinct non-zero real numbers. If the sequence satisfies the relation A_n - 5A_ n-1 + 6A_ n-2 = 0 for all integers n 3 , then the set p, q is equal to

Options

  1. A2, 3
  2. B-2, -3
  3. C1, 5
  4. D-1, -5

Correct answer

A. 2, 3

Step-by-step solution

Given the sequence A_n = 2024 p^n + 2025 q^n and the recurrence relation A_n - 5A_ n-1 + 6A_ n-2 = 0 . Substituting A_n into the recurrence relation gives: 2024 p^n + 2025 q^n - 5(2024 p^ n-1 + 2025 q^ n-1 ) + 6(2024 p^ n-2 + 2025 q^ n-2 ) = 0 Rearranging the terms, we get: 2024 p^ n-2 (p^2 - 5p + 6) + 2025 q^ n-2 (q^2 - 5q + 6) = 0 Since this equation must hold for all integers n 3 and p, q are distinct non-zero real numbers, the coefficients of p^ n-2 and q^ n-2 must independently be zero. Thus, p and q are the r

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