Recursive sequences

Task number: 2876

Show that the following recursively defined sequences \(\{a_n\}\) have limits, and determine them.

  • Variant 1

    \(a_1=\sqrt c\), where \(c\) is a positive real number, and \(a_{n+1}=\sqrt{a_n+c}\).

  • Variant 2

    \(a_1=0\) and \(a_{n+1}=a_n+\frac12 (x-a_n)^2\), for \(0\le x \le 1\).

  • Variant 3

    \(a_1=\sqrt{2}\) and \(a_{n+1}=\sqrt{2-a_n}\).

  • Variant 4

    \(a_1=1\) a \(a_{n+1}=\frac1{1+a_n}\)

  • Variant 5

    \(a_1= c\), where \(c\) is a positive real nuimber, and \(\displaystyle a_{n+1}=\frac12\left(a_n+\frac2{a_n}\right)\).

Difficulty level: Hard task
Proving or derivation task
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