Question

I need help on my Arithmetic and Geometric Sequence worksheet
-24,-8,-2 2/3, -8/9
8,-4,2,-1
3/4,-1/2,1/3,-2/9
Find the next three terms

Answer 1

Okay, do you know what an arithmetic sequence is, and a geometric sequence?

Answer 2

Arithmetic is subtracting? Geometric is multiplying

Answer 3

arithmetic is adding or subtracting, yes, and geometric is multiplying (or dividing)

An arithmetic sequence always has the same difference between two adjacent terms, and a geometric sequence always has the same quotient between two adjacent terms.

So are these arithmetic or geometric sequences?

Answer 4

Geometric sequence?

Answer 5

Yes, all 3 are geometric sequences. Can you work out what the ratio is for the first one?

\[-24,-8,-2\frac{2}{3},-\frac{8}{9}\]

Answer 6

\[\frac{-24}{-8}=\]\[\frac{-8}{-2\frac{2}{3}}=\]\[\frac{-2\frac{2}{3}}{-\frac{8}{9}}=\]

Answer 7

They all equal to 3

Answer 8

Each number in the sequence is 1/3 of the previous one, isn't it?

\[-24*\frac{1}{3} = -8\]\[-8*\frac{1}{3}=-\frac{8}{3} \ (=-2\frac{2}{3})\]\[-\frac{8}{3}*\frac{1}{3} = -\frac{8}{9}\]

Answer 9

so what is the term after \(-\dfrac{8}{9}\) going to be?

Answer 10

-8/27

Answer 11

that's right!
Here are the first 10 terms of that sequence:
\[\left\{-24,-8,-\frac{8}{3},-\frac{8}{9},-\frac{8}{27},-\frac{8}{81},-\frac{8}{243},-\frac{8}{729},-\frac{8}{2187},-\frac{8}{6561}\right\}\]

Answer 12

Now how about the next problem?
\[8,-4,2,-1\]
What is the common ratio between the terms?

Answer 13

Well the pattern is positive, negative, positive, negative. But the ratio is ... I don't know

Answer 14

Is it -2?

Answer 15

\[\frac{8}{-4} = -2\]\[\frac{-4}{2} = -2\]\[\frac{2}{-1} = -2\]
Looks like we divide by \(-2\) to get the next term...

Answer 16

That makes the next term in the sequence = ???

Answer 17

So the next 3 terms would be .5 , -.25, .125

Answer 18

Yep. Probably a bit clearer if you keep them as fractions rather than decimals.

\[8,-4,2,-1,1/2,-1/4,1/8\]

Answer 19

How about the last problem?

Answer 20

\[\frac{ 3 }{ 4 } ,-\frac{ 1 }{ 2 },\frac{ 1 }{ 3 },-\frac{ 2 }{ 9 }\]

Answer 21

Looking back at the problem, it's a little bit ambiguous as to whether we need to find the next term in each sequence, or the next 3 terms in each sequence. Probably safer to do the next 3 terms!

Answer 22

I got it!!! its \[\frac{ 1 }{ 1 }\] thats how i find the next 3 terms?

Answer 23

Nevermind

Answer 24

I'm stuck on the last one

Answer 25

\[\frac{3}{4}*\frac{a}{b} = -\frac{1}{2}\]
can you find a fraction that makes that work?

Answer 26

don't worry about the negative sign, you can just put that on afterward

\[\frac{3}{4}*\frac{a}{b} = \frac{1}{2}\]
maybe start by multiplying everything by 4
\[4*\frac{3}{4}*\frac{a}{b} = 4*\frac{1}{2}\]\[3*\frac{a}{b} = 2\]\[\frac{a}{b} = \frac{2}{3}\]
so we get the next term by multiplying by \(-\dfrac{2}{3}\)

Let's check that: we start with \frac{3}{4}:
\[\frac{3}{4}*-\frac{2}{3} = -\frac{3*2}{4*3} = -\frac{1}{2}\checkmark\]
\[-\frac{1}{2}*-\frac{2}{3} = \frac{1*2}{2*3} = \frac{1}{3}\checkmark\]
looks like that is the right multiplier...

Answer 27

Yesss

Answer 28

Thank you for helping me!! :)

Answer 29

Hopefully these will all be easy for you now!

As I mentioned earlier, you might want to play it safe and figure out the next 3 terms for each of the sequences. If it turns out you didn't need to, well, you got some practice.

Answer 30

True that