\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
We consider a string with a length of \(20\) m. Camille begins by cutting a piece measuring one-fourth of the length of the string. Then, at each subsequent step, she cuts a piece measuring one-fourth of the length of the piece cut in the previous step.
  1. Let \(u_n\) be the length cut at the \(n\)-th step.
    1. Give the value of \(u_1\) and \(u_2\).
    2. Give the expression of \(u_n\) in terms of \(n\).
    3. Determine the limit of the sequence \((u_n)\).
  2. Camille decides to glue all the cut pieces together. Let \(L_n\) be the total length of the first \(n\) pieces glued back together.
    1. Give the value of \(L_1\) and \(L_2\).
    2. Give the expression of \(L_n\) in terms of \(n\).
    3. Determine the limit of the sequence \((L_n)\).

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