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Linear AlgebraSuperchallengeDiagonalization & Matrix Powers

Diagonalization & Matrix Powers · Superchallenge problem

A sequence starts at x0=0x_0 = 0 and satisfies xn+1=2xn+3xn+4x_{n+1} = \dfrac{2x_n + 3}{x_n + 4}. Find lim⁡n→∞5n (1−xn)\displaystyle\lim_{n\to\infty} 5^n\,(1 - x_n).