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GATE 2021 IN – Question 12

Engineering Mathematics · Calculus and vector calculus · 1 mark · Multiple choice

For $x_n=0.5x_{n-1}+1$, $n=1,2,\ldots$, with $x_0=0$, the value of $\lim_{n\to\infty}x_n$ is ______.

  1. 0
  2. 1
  3. 2
  4. $\infty$

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Correct answer: (C) 2

Explanation

The fixed point L satisfies L=0.5L+1, giving L=2. Subtracting it gives xn−2=0.5(xn−1−2), so with x0=0, xn=2(1−2^(−n)). The decaying term tends to zero and the limit is **2, C**.