Question

Please help with arithmetic and geometric

Answer 1

Is this an example of an arithmetic or geometric sequence?
3,9,27,81,..

What is the common ratio or common difference of the sequence above?

Answer 2

First of all, do you understand what the arithmetic and geometric sequences are?

Answer 3

Geometric sequence.

Answer 4

Honestly no

Answer 5

An arithmetic sequence is one that increases by adding or subtracting the same value.
The geometric sequence is one that increases by multiplying or dividing the same value.

Answer 6

Im looking for help and explanations. Its my last module

Answer 7

An arithmetic sequence is a sequence with the difference between two terms constant.A geometric sequence is a sequence with the ratio between two terms.

Answer 8

Okay? So this one seems to be multiplying? So its a Geometric?

Answer 9

In this case, if you notice what is going on between the values in the sequence;
3 to 9
We are either adding by 6 or multiplying by 3. We can check the next consecutive sequence members to figure out which:
9 to 27
Adding 6 to 9 just makes 15, so it isn't addition / arithmetic. We can, however, multiply by 3 and get 27. Therefore, this sequence is geometric.

Answer 10

Arithmetic sequence Example:
2, 4, 6, 8, 10...

Answer 11

Oh okay thank you. But what about the other part?

Answer 12

The common ratio/difference is the value we multiply/add in each case, respectively.
Each time, we are multiplying by 3. So our common ratio (multiplication here) is 3.

Answer 13

Okay. Can you help me with 3 other questions?

Answer 14

You may post them and we can see what we can do. :)

Answer 15

Heres the next one---- Rewrite the following sequence, so that it represents a series rather than a sequence.
3,9,27,81

Answer 16

So, do you know what a series is? Or have you seen summation notation like this:
\( \displaystyle \sum _{k=1}^{4} a_k \)

Answer 17

Nope to both

Answer 18

Hm.. a series is just the sum of all the terms in a sequence.
So if you have not seen the sum notation, I think we are just trying to write it as the addition of each number in the sequence:
3, 9, 27, 81: <-- numbers in the sequence
3 + 9 + 27 + 81 <--- we add them together here
Sum notation makes it more conveniently expressed. Because each member of the sequence is identified by how many times you multiply by 3, the sequence could be rewritten:
3^1, 3^2, 3^3, 3^4
In general, the k-th term is 3^k. So the series would be:
\( \displaystyle \sum_{k=1}^{4} 3^k = \underbrace{3^1}_{k=1} + \underbrace{3^2}_{k=2} + 3^3 + 3^4 \)

Answer 19

Okay I think I follow you so far

Answer 20

I think for learning series you would be expected to know summation notation (the big sigma), but I don't know what is expected for your class.
The series would be: 3 + 9 + 27 + 81 or 3^1 + 3^2 + 3^3 + 3^4, or the \(\Sigma\) 3^k, all are the same representation of a series of that sequence, I am thinking..

Answer 21

Okay I understand now... The next one is ---- Find the sum of the geometric sequence 2,4,8,... if there are 20 terms.

Answer 22

Do you have a formula for geometric series or partial sum?

Answer 23

We could go through the motion of multiplying out each term, then adding them all together... but the formula should require a bit less work.

Answer 24

Do you mean this thing-- an=a1r^(n-1) **the n and 1 and below the a's

Answer 25

That is for geometric sequence, but we need to determine the series.
If I remember right, it looked like this:
sum from 1 to n of a*r^n = a*(1 - r^(n-1)) / (1 - r)

Answer 26

Bu the question says find the sum of the geometric sequence?

Answer 27

The one for the series looks like this--- Sn= a1- a1r^n/ 1-r

Answer 28

Series is the sum of a sequence.
We need to add up all these terms:
2, 4, 8, 16, 32, 64, 128, 256, ... <-- just keeps multiplying by 2
So the formula is just a derived tool so that we don't need to calculate every number in the sequence, we just use our initial information and the common ratio and... the formula pops out an answer!
Yup, that looks right.
\( S_n = \dfrac{a_1 - a_1 r^n}{1 - r} \)
a1 is the first term, r is the common ratio.

Answer 29

** and n is the number of terms we are adding up.

Answer 30

Okay I think

Answer 31

So your first term in the sequence 2, 4, 8, 16, ... is the 2.
The common ratio is what we multiply by in each step, which is also 2.
The number of terms was what we are asked for, 20 terms.
Use our formula:
\( S_{20} = \dfrac{\color{green}2 - \color{green}2 \times \color{blue}2^{20}}{1 - \color{blue}2} \)

Answer 32

Is the answer -2^20/-1

Answer 33

And would the next 2 terms be put into this formula too?

Answer 34

Nope, this is the whole sum the formula is dealing with. The sum of all 20 terms in the sequence IS what the formula outputs, and it only requires the first term of the series.

Answer 35

Im confused. So the answer is not what I wrote? Im running out of time. I have to finish at 6:30 and I still have one more problem :(

Answer 36

(2 - 2*2^20)/(1 - 2) = (2 - 2^21)/(-1)
Can you follow that?
We don't combine the 2 - 2, just multiply 2^20 by 2 which is equivalent of 2^21.

Answer 37

And then use a calculator to evaluate.

Answer 38

Umm okay and that will give me the final answer?

Answer 39

Yes.

Answer 40

Okay last question-- What is the sum of the arithmetic sequence 9,14,19,... if there are 34 terms?

Answer 41

Can you please solve this one for me? Im out of time and my teachers telling me to hurry and finish up

Answer 42

Calculate the 34th term: a_n = a_1 + n d a_1 is the first term, n = 34, and d is common difference of 5.
Then use the formula:
S_n = (n (a_1 + a_n)/2

Answer 43

a_34 = 9 + 34 * 5
S_34 = (34 * (9 + a_34)) / 2

Answer 44

Okay is that the answer?

Answer 45

If you calculate it out, yes, that gives the sum of the arithmetic sequence.

Answer 46

Okay thank you so very much :)

Answer 47

One moment.
a_34 = 9 + 5*(34 - 1)

The arithmetic sum lines up that way because the first term is for n=0.
My mistake, but it is harder to operate with the pressure. >.<

Answer 48

So a_34 = 9 + 5*33 = 174 <<< rather than 179 that comes from the original formula
The other formula is correct.

Answer 49

Okay thank you. I got an A

Answer 50

Good work! And I'm glad to help. :)

Answer 51

Destiny how are you?