A resursive sequence

  1. We define $a_{1}=-1$ and for each $n\geq 1,$we have MATHDetermine if the sequence $\{a_{n}\}$converges or diverges.

    [Sol]. We first try out some few terms. [We use MATH as the iteration function.] We obtain the following:
    $.\allowbreak $We conjecture that this sequence converges and say it converges to $a,$then we are about to solve an equation like this: MATHThe solution are : MATH. Since initial value MATH the limit should be $a=\sqrt{2}.$Next, we can use 'fixed method' to see if we have a solution for $y=x$ and $y=f(x)$ graphically.
    Indeed, we do see two intersections. We conclude that the sequence is convergent and its limit is $\sqrt{2}.$

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