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The sequence -12, -8.5, -5, -1.5, ...converges to 0decreases over its domaindivergesconverges to -1

1 Answer

4 votes

The sequence can be found using an arithmetic sequence:


a_n=a_1+(n-1)d

Where:

a1 = First term

d = Common difference

so:


\begin{gathered} a_2=-8.5=-12+(2-1)d \\ -8.5=-12+d \\ d=12-8.5 \\ d=3.5 \end{gathered}

Therefore, the sequence is given by:


\begin{gathered} a_n=-12+(n-1)\cdot3.5 \\ a_n=3.5n-15.5 \end{gathered}

Let's find the limit of the function associated to an:


\begin{gathered} \lim _(n\to\infty)a_n=\lim _(x\to\infty)f(x) \\ where \\ f(x)=3.5x-15.5 \\ so\colon \\ \lim _(x\to\infty)(3.5x-15.5)=\lim _(x\to\infty)3.5x=3.5\lim _(x\to\infty)x=3.5\cdot\infty=\infty \end{gathered}

Therefore, since the limit of the function associated to the sequence tends to infinity, we can conclude that the sequence diverges

User Mike Pierce
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