MHB Can't solve a sequence (to determine if a given value is a member)

AI Thread Summary
The discussion revolves around determining if the fraction 41/81 is a member of the sequence defined by a_n = (n^2 + 1) / (2n^2). Participants guide the original poster through the process of cross-multiplying and simplifying the equation to find a natural number solution for n. After deriving the equation 81(n^2 + 1) = 82n^2, they confirm that n equals 9, indicating that 41/81 is indeed the 9th element of the sequence. The conversation also touches on exploring the monotonicity of the sequence and its graphical representation, as well as distinguishing between arithmetic and geometric sequences. Ultimately, the original poster expresses gratitude for the assistance and seeks further clarification on related mathematical concepts.
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Can somebody help me with this problem? I have to proof if 41/81 is a part of attached sequence. Step by step guide would be useful. Thank you! http://img.tapatalk.com/d/13/10/18/2unyje8y.jpg
 
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Re: Cant solve a sequence

andreask said:
Can somebody help me with this problem? I have to proof if 41/81 is a part of attached sequence. Step by step guide would be useful. Thank you!

a_n = \frac{n^2 +1}{2n^2}

\frac{n^2 +1}{2n^2} = \frac{n^2}{2n^2} + \frac{1}{2n^2} = \frac{1}{2} + \frac{1}{2n^2}

can we find "n" such that

\frac{1}{2} + \frac{1}{2n^2} = \frac{41}{81} ?
solve it for n
 
Re: Cant solve a sequence

Amer said:
a_n = \frac{n^2 +1}{2n^2}

\frac{n^2 +1}{2n^2} = \frac{n^2}{2n^2} + \frac{1}{2n^2} = \frac{1}{2} + \frac{1}{2n^2}

can we find "n" such that

\frac{1}{2} + \frac{1}{2n^2} = \frac{41}{81} ?
solve it for n

thank you,but i still don't get it...
 
Re: Cant solve a sequence

I would write:

$$\frac{n^2+1}{2n^2}=\frac{41}{81}$$

Now, cross-multiply and then see if you can find a natural number solution.
 
Re: Cant solve a sequence

MarkFL said:
I would write:

$$\frac{n^2+1}{2n^2}=\frac{41}{81}$$

Now, cross-multiply and then see if you can find a natural number solution.

Can you continiue?
And many thanks
 
Re: Cant solve a sequence

andreask said:
Can you continiue?
And many thanks

I would rather help you continue, so that you learn more. :D

What do you get when you cross-multiply? If you are unfamiliar with this method, if you have:

$$\frac{a}{b}=\frac{c}{d}$$

then cross-multiplying, which means to multiply the numerators on each side by the denominators on the other, and then equating both products, will give you:

$$ad=bc$$

It is a kind of shortcut for multiplying both sides by the common denominator of $bd$:

$$\frac{a}{b}bd=\frac{c}{d}bd$$

Canceling, or reducing the fractions, we get:

$$ad=bc$$

Can you do this with the equation I gave?
 
Re: Cant solve a sequence

MarkFL said:
I would rather help you continue, so that you learn more. :D

What do you get when you cross-multiply? If you are unfamiliar with this method, if you have:

$$\frac{a}{b}=\frac{c}{d}$$

then cross-multiplying, which means to multiply the numerators on each side by the denominators on the other, and then equating both products, will give you:

$$ad=bc$$

If it a kind of shortcut for multiplying both sides by the common denominator of bd:

$$\frac{a}{b}bd=\frac{c}{d}bd$$

Canceling, or reducing the fractions, we get:

$$ad=bc$$

Can you do this with the equation I gave?
Yes,i understand this!

So i think that's my result: 81*(n^2+1) = 82(n^2)
So 41/81 is not a member of my sequence. Am i right?
I would love to learn more about such sequences,do you know some good reference?
And again,thank you!
 
Re: Cant solve a sequence

You have the correct equation:

$$81\left(n^2+1 \right)=41\left(2n^2 \right)$$

Now, distribute on both sides, then see if you can solve for $n$...

And no, I do not know of any online resources for studying sequences specifically.
 
Re: Cant solve a sequence

MarkFL said:
You have the correct equation:

$$81\left(n^2+1 \right)=41\left(2n^2 \right)$$

Now, distribute on both sides, then see if you can solve for $n$...

And no, I do not know of any online resources for studying sequences specifically.

81n2+1 = 82n2
1 = 82n2-81nn
1= n2

I don't know i I am right...or what to do...
 
  • #10
Re: Cant solve a sequence

andreask said:
81n2+1 = 82n2
1 = 82n2-81nn
1= n2

I don't know i I am right...or what to do...

When you distribute the $81$ on the left side, you need to do so to both terms within the parentheses, as follows:

$$81n^2+81=82n^2$$

Next, subtract $81n^2$ from both sides, and what are we left with?
 
  • #11
Re: Cant solve a sequence

MarkFL said:
When you distribute the $81$ on the left side, you need to do so to both terms within the parentheses, as follows:

$$81n^2+81=82n^2$$

Next, subtract $81n^2$ from both sides, and what are we left with?
81 = n2 ??
 
  • #12
Re: Cant solve a sequence

andreask said:
81 = n2 ??

Yes, good! :D

So, what is the positive root? What is the positive value of $n$?
 
  • #13
Re: Cant solve a sequence

MarkFL said:
Yes, good! :D

So, what is the positive root? What is the positive value of $n$?

9 of course.

But what have i proved?That 41/81 is a 9th element of sequence?
 
  • #14
Re: Cant solve a sequence

andreask said:
9 of course.

But what have i proved?That 41/81 is a 9th element of sequence?

Yes, you have shown that:

$$a_{9}=\frac{41}{81}$$
 
  • #15
Re: Cant solve a sequence

MarkFL said:
Yes, you have shown that:

$$a_{9}=\frac{41}{81}$$

Ok,thank you.

i still have three questions:

1. How can i explore the monotonoty of this sequence?
2. How to draw this sequence in coordinate system?
3. Can you show me another sequence solution(similar to mine)? Of course,if you have time...
 
  • #16
Without using calculus, to explore the monotonicity of the sequence, I would write the $n$th term as suggested above by Amer:

$$a_{n}=\frac{1}{2}+\frac{1}{2n^2}$$

Can you see that each succeeding term is smaller than the last, and that as $n$ grows without bound $a_{n}$ asymptotically approaches $$\frac{1}{2}$$?

You could plot the first few terms, along with the asymptote.

You could make up another such problem just as easily as I can. :D
 
  • #17
MarkFL said:
Without using calculus, to explore the monotonicity of the sequence, I would write the $n$th term as suggested above by Amer:

$$a_{n}=\frac{1}{2}+\frac{1}{2n^2}$$

Can you see that each succeeding term is smaller than the last, and that as $n$ grows without bound $a_{n}$ asymptotically approaches $$\frac{1}{2}$$?

You could plot the first few terms, along with the asymptote.

You could make up another such problem just as easily as I can. :D
Seems I am to stupid for sequences...
 
  • #18
andreask said:
Seems I am to stupid for sequences...

No, that's not the case at all...let's look at the $n$th term:

$$a_{n}=\frac{1}{2}+\frac{1}{2n^2}$$

And let's for now just look at this part of the $n$th term:

$$\frac{1}{2n^2}$$

Would you agree that when the numerator of a fraction is fixed, that is remains constant, and the denominator gets bigger and bigger, then the value of the fraction is getting smaller and smaller?
 
  • #19
MarkFL said:
No, that's not the case at all...let's look at the $n$th term:

$$a_{n}=\frac{1}{2}+\frac{1}{2n^2}$$

And let's for now just look at this part of the $n$th term:

$$\frac{1}{2n^2}$$

Would you agree that when the numerator of a fraction is fixed, that is remains constant, and the denominator gets bigger and bigger, then the value of the fraction is getting smaller and smaller?

Yes,i agree!
I think that this sequence is limited too 0,5, but i don't know how to prove it...i have started a master degree with 38,my math was very good at high school,but now...now is all at higher level,i really don't know how would i master this all things...
 
  • #20
andreask said:
Yes,i agree!
I think that this sequence is limited too 0,5, but i don't know how to prove it...i have started a master degree with 38,my math was very good at high school,but now...now is all at higher level,i really don't know how would i master this all things...

What does $$\frac{1}{2n^2}$$ approach as $n$ grows without bound?
 
  • #21
MarkFL said:
What does $$\frac{1}{2n^2}$$ approach as $n$ grows without bound?

1/2 ?

I really don't know how to get it...

And i really apreciate that you have time for me,thank you
 
  • #22
andreask said:
1/2 ?

I really don't know how to get it...

And i really apreciate that you have time for me,thank you

Let's look at the first few term of this fraction...let:

$$b_{n}=\frac{1}{2n^2}$$

We find:

$$b_1=\frac{1}{2(1)^2}=\frac{1}{2}$$

$$b_2=\frac{1}{2(2)^2}=\frac{1}{8}$$

$$b_3=\frac{1}{2(3)^2}=\frac{1}{18}$$

$$b_4=\frac{1}{2(4)^2}=\frac{1}{32}$$

$$\vdots$$

$$b_{1000}=\frac{1}{2(1000)^2}=\frac{1}{2000000}$$

$$\vdots$$

$$b_{1000000}=\frac{1}{2(1000000)^2}=\frac{1}{2000000000000}$$

Do you see that as $n$ gets bigger and bigger, $b_{n}$ gets smaller and smaller? If we could let $n$ go to infinity, then what would $b_{n}$ be?
 
  • #23
MarkFL said:
Let's look at the first few term of this fraction...let:

$$b_{n}=\frac{1}{2n^2}$$

We find:

$$b_1=\frac{1}{2(1)^2}=\frac{1}{2}$$

$$b_2=\frac{1}{2(2)^2}=\frac{1}{8}$$

$$b_3=\frac{1}{2(3)^2}=\frac{1}{18}$$

$$b_4=\frac{1}{2(4)^2}=\frac{1}{32}$$

$$\vdots$$

$$b_{1000}=\frac{1}{2(1000)^2}=\frac{1}{2000000}$$

$$\vdots$$

$$b_{1000000}=\frac{1}{2(1000000)^2}=\frac{1}{2000000000000}$$

Do you see that as $n$ gets bigger and bigger, $b_{n}$ gets smaller and smaller? If we could let $n$ go to infinity, then what would $b_{n}$ be?

I think that bn would be zero,or very close.
but beacuse of + 1/2 bn can't be smaler then 0,5. Am i right?

And sorry,i have to learn to write fractions etc. in right form.
 
  • #24
Yes, $b_{n}$ gets closer and closer to zero, and since:

$$a_n=\frac{1}{2}+b_n$$

then $$a_{n}$$ gets closer and closer to $$\frac{1}{2}$$.

Does this make sense?
 
  • #25
MarkFL said:
Yes, $b_{n}$ gets closer and closer to zero, and since:

$$a_n=\frac{1}{2}+b_n$$

then $$a_{n}$$ gets closer and closer to $$\frac{1}{2}$$.

Does this make sense?

Yes,it makes sense ,of course. Only problem si,how to write this in a proper way for my profesor? :pAnd why is $$\frac{n^2+1}{2n^2}$$ equal to $$\frac{1}{2} + \frac{1}{2n^2}$$ ?
 
  • #26
andreask said:
Yes,it makes sense ,of course. Only problem si,how to write this in a proper way for my profesor? :pAnd why is $$\frac{n^2+1}{2n^2}$$ equal to $$\frac{1}{2} + \frac{1}{2n^2}$$ ?

A property of fractions is that:

$$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$

so that:

$$\frac{n^2+1}{2n^2}=\frac{n^2}{2n^2}+\frac{1}{2n^2}=\frac{1}{2}+\frac{1}{2n^2}$$
 
  • #27
MarkFL said:
A property of fractions is that:

$$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$

so that:

$$\frac{n^2+1}{2n^2}=\frac{n^2}{2n^2}+\frac{1}{2n^2}=\frac{1}{2}+\frac{1}{2n^2}$$

Now i see...and understand.

Only thing that's left,how to write it on the paper,so that would be acteptable for my teacher...

I owe you a drink now :D
 
  • #28
From what course is this problem? If it is for a course prior to calculus, then you will simply need to explain it as best you can in your own words. You could also choose to write the $n$th term as:

$$a_{n}=\frac{1+\frac{1}{n^2}}{2}$$

Now it is easy to see that the denominator is fixed, and the numerator is getting smaller and smaller as $n$ gets larger and larger, that is, the sequence is strictly decreasing, i.e., it is monotonic.
 
  • #29
MarkFL said:
From what course is this problem? If it is for a course prior to calculus, then you will simply need to explain it as best you can in your own words. You could also choose to write the $n$th term as:

$$a_{n}=\frac{1+\frac{1}{n^2}}{2}$$

Now it is easy to see that the denominator is fixed, and the numerator is getting smaller and smaller as $n$ gets larger and larger, that is, the sequence is strictly decreasing, i.e., it is monotonic.

Its prior to algebra...

I will try to do somehow...
 
  • #30
I would think the argument I gave in my post previous to this would suffice then.
 
  • #31
MarkFL said:
I would think the argument I gave in my post previous to this would suffice then.

I agree...

But,now i have new problem,about Mathematical induction,should i open new Thread?
 
  • #32
andreask said:
I agree...

But,now i have new problem,about Mathematical induction,should i open new Thread?

Yes, and I would post it in our Discrete Mathematics sub-forum.
 
  • #33
MarkFL said:
Yes, and I would post it in our Discrete Mathematics sub-forum.

I will...

Maybe you can take a look...if you want...
 
  • #34
I still have one question...What kind of type this sequence is? Arithmetic or geometric? I can't figure it out...
 
  • #35
andreask said:
I still have one question...What kind of type this sequence is? Arithmetic or geometric? I can't figure it out...

An arithmetic sequence has the property:

$$a_{n+1}-a_{n}=d$$

While a geometric sequence would have the property:

$$\frac{a_{n+1}}{a_{n}}=r$$

Note: Both $d$ and $r$ are constants that do not depend on $n$. Does this sequence satisfy either property?
 
  • #36
MarkFL said:
An arithmetic sequence has the property:

$$a_{n+1}-a_{n}=d$$

While a geometric sequence would have the property:

$$\frac{a_{n+1}}{a_{n}}=r$$

Note: Both $d$ and $r$ are constants that do not depend on $n$. Does this sequence satisfy either property?

It seems its arthmetic
 
  • #37
andreask said:
It seems its arthmetic

I find:

$$a_{n+1}-a_{n}=\frac{(n+1)^2+1}{2(n+1)^2}-\frac{n^2+1}{2n^2}=-\frac{2n+1}{2n^2(n+1)^2}$$

Thus, the difference between two succeeding terms is a function of $n$, and so the sequence is not arithmetic.
 
  • #38
MarkFL said:
I find:

$$a_{n+1}-a_{n}=\frac{(n+1)^2+1}{2(n+1)^2}-\frac{n^2+1}{2n^2}=-\frac{2n+1}{2n^2(n+1)^2}$$

Thus, the difference between two succeeding terms is a function of $n$, and so the sequence is not arithmetic.

Ad its not geometric?
 
  • #39
andreask said:
And its not geometric?

What do you find when you compute the ratio I gave above?
 
  • #40
MarkFL said:
What do you find when you compute the ratio I gave above?

Will try tomorrow,im litlle busy now
 

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