Question

Geometric Sequence help please!!!

What is the first term of the geometric sequence presented in the table below?

n 3 6
an 12 -324

@peachpi I hope its okay that I tagged you. I know I have to solve for a1, and I'm pretty sure I can find r by y2-y1, but I don't know where to go from there.

Answer 1

Let a1 be the first term of the sequence, and an be the nth term.

Then, for any integer n equal to 0 or greater, an=a1*r^n.

You are given two values of the sequence along with the pertinent n values. Write out the above equation twice, once for n=3 and once for n=6.

Look for a way to solve the results for the first term, a1.

Answer 2

I need to find the value of r first right? Is it -27?

Answer 3

You cannot find r through subtraction, because this is a geometric series, not an arithmetic series.

Just supposing that r were -27: substitute that into an=a1*r^n. can you find a possible value for a1? You might need to experiment a bit to find a1 and r that suit this problem situation.

Answer 4

There's an example in my notes that defines r from the table
n 1 2
an 6 -3

They don't give a formula but they put -3/6 and then say r=-1/2. I'm just going off the one example I have, and it worked for the 4 other problems I had to do for this assignment.

I realize that I made a mistake when I asked the question. I meant to say y2/y1.

Answer 5

Do I need a1 to define r? Or can I do it from any point?

Answer 6

You may determine a1 and r in either order. Your goal is to find two equations based upon the 2 sequence values given and their associated n, and then to solve these equations simultaneously. I haven't finished check ing my own work, but have arrived at a tentative value for a1 of -4/9.

Just supposing that a1 actually does equal -4/9. could you then find r? Try it.

Then see wehther your equation \[a _{n}=a _{1}r^n\] correctly predicts the sequence values for n=3 and n=6.

Answer 7

Have to be willing to experiment. I've tried this problem several times and have only just now come up with a1 and r values that seem to predict the sequence values for n=3 and n=6.

Answer 8

Quoting you:

"There's an example in my notes that defines r from the table
n 1 2
an 6 -3

They don't give a formula but they put -3/6 and then say r=-1/2. I'm just going off the one example I have, and it worked for the 4 other problems I had to do for this assignment." Right; they don't give you a formula. So, you have license to adopt your own model for the formula defining a geom. sequence. Mine, for this problem, is
\[a _{n}=a _{1}*r^n\]

Answer 9

12=(-4/9)r^(3-1)
12=(-4/9)r^2

I'm using a basic calculator and I can't simplify more than this. Also the answer choices for a1 are 4/3, -3/4, -3, and 3/4. Did I substitute the values in correctly though?

Answer 10

Sounds as tho you are focusing on n=3 here, given that your sequence value is 12.

Answer 11

Your model is different from mine in one respect: you use r^(n-1), whereas I use r^n.
Your model would begin with n=1, whereas my model would begin with n=0.

The test of the pudding is whether or not your formula correctly predicts \[a _{6}=-324. \]

Answer 12

What value do you believe a1 has, and why?
what value do you believe r has, and why?

Answer 13

Just because our formulas differ a bit does NOT mean that one is wrong and the other right.

Answer 14

I don't know. I've been working on this problem for an hour and I've gone through two sheets of paper plugging in values. I checked through my notes and on a different page they gave me an=ar^n-1. I don't doubt that you are right, I just don't know how to get an answer based off the formula that I am expected to use.

Answer 15

Earlier, you typed the following:

12=(-4/9)r^(3-1)
12=(-4/9)r^2

This tells me that you are testing the possibility that a1=-4/9, and that you prefer to use the notation (-4/9)r^(n-1). This is OK!

We are told that if n=6, the 6th term of the sequence is -324.

Writing this out, with the purpose of trying to determine r,

-324 = (-4/9)*r^(6-1).

Here the only unknown is r, so you should be able to find r.

My results are a1=-4/9 and r=-3.

Answer 16

hello

Answer 17

What's the best thing I could do at this point to help you? I, too, have 2 pages worth of scribbling done as I experimented. But I've come up with those a1 and r values and have checked them and found them to be correct. This work becomes easier with practice.

Answer 18

My friend who's helping me randomly decided that r=3 and got one of my options for an answer. Could r be 3? I would have to plug it into two equations to see if I get the answer of -3/4 for a1 in both scenarios.

12=a(3)^(3-1)
12=9a
a=4/3

in the scenario she worked out when r=3 she used the other set of points

Answer 19

I applaud this approach, even though I don't agree with the value of r your friend got.

Despite my assurances that you could use either version of the geom. sequence formula, I'm going to ask you now to use mine:\[a _{n}=a _{1}*r^n\]

Answer 20

We know that if n=3, a(3) = 12. Let's write that out: 12=a1*r^3. Can you agree with that?

Answer 21

Yes that makes sense

Answer 22

It states that when n=3, the corresponding term of this geom sequence is 12.

Squaring everything results in (12)^2=(a1)^2*r^6.

Answer 23

Okay. I think I'm following.

Answer 24

Now, if n=6, we get a(6)=-324*r^6. But from the previous equation, we have
144/(a1)^2 = r^6, so we can substitute this for r^6 in the previous equation, thus eliminating r and allowing us to focus on a1 alone. Agreed?

Answer 25

First equation was\[12=a _{1}r^6\]

Answer 26

Second equation gave us\[\frac{ 144 }{ a _{1}^2 }=r^6\]

Answer 27

If you agree, eliminate r^6 by replacing r^6 in the first equation by \[\frac{ 144 }{ a _{1}^2 }\]

Answer 28

Stop me if you're not comfortable with anything here.

Answer 29

This all makes sense but I don't think I could replicate it on a similar problem.

Answer 30

I have 50+ years worth of experience doing this, so obviously have had some practice. With practice, you'd do just fine.

Answer 31

If y ou pursue this line of reasoning, you could show that a1=-4/9.

Now, subbing a1=-4/9 into the first equation,

Answer 32

Well thank you for taking so much time to help me with this problem. The other ones in the assignment I did in about 5 minutes total. I don't know why this one is so much more complicated.

Answer 33

complicated partially because you have to settle on a model equation to use for this geom series, and partially because you have to solve for 2 constants, a1 and r.

Are you fully satisfied here, or is there anything else you need to do before we split?

Answer 34

I've experimented and have found that the model an=a1*r^n works best for me.

Answer 35

No. Thank you so much!

Answer 36

You're very welcome. You're a pleasure to have around!