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Sequences and Series

Sequences and Series: Level 3 Challenges


If x3,x,x2x^3, x, x^2 form an arithmetic progression (in the given order), find the sum of all possible values of xx.

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

In the answer options, ϕ=1+52 \phi = \frac{ 1 + \sqrt{5} } { 2} (the golden ratio).

Cody has started running in a well-organized manner. He runs 100 m100 \text{ m} east, then turns left and runs another 10 m10 \text{ m} north, turns left and runs 1 m,1 \text{ m}, again turns left and runs 0.1 m,0.1 \text{ m}, and on the next turn 0.01 m,0.01 \text{ m}, and so on.

Assuming that Cody can run in this pattern infinitely, the displacement from his initial position can be written as ab\frac{a}{\sqrt{b}} with aa and bb being positive integers and bb square-free.

What is the value of a×b? a \times b?

101(1+0.1+0.01+0.001+)+102(2+0.2+0.02+0.002+)+103(3+0.3+0.03+0.003+)+...\begin{aligned} 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{aligned}

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

Clarification: In the nthn^\text{th} term, after the string of n1n - 1 zeroes, the number under the bar is nn.

If the sum of an arithmetic progression of six positive integer terms is 78, what is the greatest possible difference between consecutive terms?

24816=?\Large \sqrt{2 \sqrt{4 \sqrt{8 \sqrt{16\ldots}}} } = \, ?


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