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A1=12−12,a2=23−13,a3=34−14,a4=45−15,a5=56−16. a1=12−12,a2=23−13,a3=34−14,a4=45−15,a5=56−16.
a.find an explicit formula for anan: .
b.determine whether the sequence is convergent or divergent: . (enter "convergent" or "divergent" as appropriate.)
c.if it converges, find limn→∞an=limn→∞an= .

Respuesta :

LammettHash
[tex]a_1=12-12=(1\times12-0)-(12+0)[/tex]
[tex]a_2=23-13=(2\times12-1)-(12+1)[/tex]
[tex]a_3=34-14=(3\times12-2)-(12+2)[/tex]
[tex]a_4=45-15=(4\times12-3)-(12+3)[/tex]
[tex]a_5=56-16=(5\times12-4)-(12+4)[/tex]

Clearly, the general for the [tex]n[/tex]-th term is

[tex]a_n=(12n-(n-1))-(12+n-1)=10n-10[/tex]

(Aside: given the simple pattern, it's curious why the terms would be given the way they are. It's easier to divine [tex]10n-10[/tex] from [tex]a_1=0,a_2=10,a_3=20,\ldots[/tex], but that's just my opinion.)

The sequence does not converge. As [tex]n\to\infty[/tex], so does [tex]a_n\to\infty[/tex].

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