Discussion
If \(x_{1} = 1\), \(x_{2} = 1 + \frac{1}{2}\), \(x_{3} = 1 + \frac{1}{2} + \frac{1}{2^{2}}\), ......., \(x_{n} = 1 + \frac{1}{2} + \frac{1}{2^{2}} + \cdots + \frac{1}{2^{n-1}}\), then which of the following statements is true ?
Answer: D
The true statement is \(x_{n} \lt 2\) for every n.
\(x_{n}\) is the sum of the first n terms of a geometric series with first term 1 and ratio \(\frac{1}{2}\):
\[ \begin{aligned} x_{n} &= \frac{1 - (1/2)^{n}}{1 - 1/2} \\ &= 2 - \frac{1}{2^{n-1}} \end{aligned} \]
Since \(\frac{1}{2^{n-1}} \gt 0\) for every n, \(x_{n}\) is always less than 2, though it gets as close to 2 as we like.
So \(x_{n}\) never equals 2 and never exceeds 2, which rules out the options '\(x_{n} = 2\) for some n' and '\(2 \lt x_{n} \lt 3\)'. And \(x_{4} = 1.875\), already above 1.8, so '\(x_{n} \le 1.8\) for every n' is false.
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