What's the sum of the first 100 positive integers? How about the first 1000?

If \(x^3, x, x^2\) form an arithmetic progression (in the given order), find the sum of all possible values of \(x\).

**Clarification**: An arithmetic progression is allowed to be a constant sequence.

Cody has started running in a well organized manner. He runs \(100 \text{ m}\) east, then turns left and runs another \(10 \text{ m}\) north, turns left and runs \(1 \text{ m},\) again turns left and runs \(0.1 \text{ m},\) and on next turn \(0.01 \text{ m}\) and so on. Assuming that Cody can run in this pattern infinitely, then the distance from his initial position can be written as \(\frac{a}{\sqrt{b}}\) with \(a\) and \(b\) as positive integers and \(b\) square-free.

What is the value of \( a \times b?\)

\[\begin{align*} 10^{-1}(1 + 0.1 + 0.01& + 0.001 + \cdots) + \\ 10^{-2}(2 + 0.2 + 0.02& + 0.002 + \cdots) + \\ 10^{-3}(3 + 0.3 + 0.03& + 0.003 + \cdots) + \\ &. \\ &. \\ &. \\ \end{align*}\]

The above sum can be represented as \(\left(\dfrac{a}{b}\right)^{2}\), where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b\).

**Clarification**: In the \(n^\text{th}\) term, after the string of \(n - 1\) zeroes, the number under the bar is \(n\).

\[\Large \sqrt{2 \sqrt{4 \sqrt{8 \sqrt{16\ldots}}} } = \, ? \]

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