Question

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= .

Answer 1

`a_1=12-12=(1*12-0)-(12+0)`

`a_2=23-13=(2*12-1)-(12+1)`

`a_3=34-14=(3*12-2)-(12+2)`

`a_4=45-15=(4*12-3)-(12+3)`

`a_5=56-16=(5*12-4)-(12+4)`

Clearly, the general for the
`n`-th term is


`a_n=(12n-(n-1))-(12+n-1)=10n-10`

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

The sequence does not converge. As
`n\to\infty`, so does
`a_n\to\infty`.