Question

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

Answer 1

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