Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

72, 18, 9/2 , 9/8 , ...

Converges; 0

Converges; -6120

Diverges

Converges; 96

Answer 1

First off,what kind of sequence is this?

Answer 2

I have no clue

Answer 3

This is what's called a "geometric" sequence.

What that means is that if you divide the second term by the first, the third term by the second, the fourth term by the third, etc...

You should get the same quotient.

Answer 4

oh ok cool

Answer 5

It is impossible to tell convergence without a general term. Four terms does not a definition of in infinite sequence make. The 5th terms may be -2. Where would that leave us?

Answer 6

@tkhunny lol that is true :3
-It is always possible to find a rule which justifies a finite sequence to be continued by any number-
--The Oxford Murders


But let's stick to the simplest, whatever that means :D

Answer 7

@tholn
Divide the second term by the first, what do you get?

Answer 8

72/18=4

Answer 9

Two things:

1) The author of the question should know better, or

2) You are expected to point out this deficiency and then solve the problem with whatever assumptions seem reasonable, such as the fact that the first three ratios are the same.

Answer 10

divide the second term by the first... means second over first
@tholn a bit... backwards, lol What I meant would be
18/72 = 1/4

everything all right so far?

Answer 11

yes

Answer 12

Okay, now divide the third term by the second...

Answer 13

(9/2)/18 = 1/4

Answer 14

And finally, divide the fourth term by the third (sorry if this is tedious, this'll be the last division, I promise :) )

Answer 15

(9/8)/(9/2)=1/4

Answer 16

Okay. You'll notice that every time we divided (second over first, third over second, fourth over third, etc)

The quotient is 1/4

1/4 is what you call the "common ratio" of the geometric sequence.

Now if the common ratio is STRICTLY BETWEEN -1 and 1, then the sequence ALWAYS converges to zero :)

IS 1/4 between -1 and 1?

\[\Large -1 < \frac14 <1 \qquad\qquad\color{red}{??}\]

Answer 17

oh ok so it Converges; 0

Answer 18

That is correct ;)

Answer 19

Also note, that if you get a common ratio that it -1 or less , or 1 or more, in other words, if you get a common ratio r such that...
\[\Huge |r|\ge 1\]

Then the sequence ALWAYS diverges.

Answer 20

u explain sooo well!