GATE 2025 BT – Question 59
Let $a_0 = 0$ and define $a_n = \frac{1}{2}(1 + a_{n-1})$ for all positive integers $n \ge 1$. The least value of $n$ for which $|1 - a_n| < \frac{1}{2^{10}}$ is ______. (Answer in integer)
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Correct answer: 11
Explanation
$1 - a_n = 1 - \frac{1 + a_{n-1}}{2} = \frac{1 - a_{n-1}}{2}$, so $1 - a_n = \frac{1}{2^n}$ (since $1 - a_0 = 1$). The condition $\frac{1}{2^n} < \frac{1}{2^{10}}$ needs $n > 10$, so the least $n$ is 11.