Math Is Fun Forum

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Geometric Series

If a/b rounded to the nearest trillionth is 0.008012018027, where a and b are positive integers, what is the smallest possible value of a+b?

How do I do this problem? Thanks.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

A sequence of real numbers (x_n) is defined recursively as follows: x_0=a and x_1=b are positive real numbers, and
x_(n+2) = (x_(n+1)+1)/(x_n)
for n = 0, 1, 2,.... Find the value of x_{2012}, in terms of a and b.

I've tried finding a pattern, but the calculations get messy pretty quickly. Help?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

I still would have liked to have seen your work. Without it, I have no idea what help you need. Since you are in the thread Help Me I would think that is your main concern.

I've tried finding a pattern, but the calculations get messy pretty quickly

Let me see that attempt at a pattern. The answer is not difficult at all.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Geometric Series

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geometric Series

Hello:

@Bobby: solving the second question was easy enough but how did you do the first?

Last edited by ElainaVW (2016-08-12 18:40:45)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geometric Series

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

@bobby
I made a mistake simplifying one of the fractions in Problem 2...Thanks.

3.
Let
A_0 = 6
A_1 = 5
A_n = A_{n - 1} + A_{n - 2} for n ≥ 2
There is a unique ordered pair (c,d) such that c*phi^n + d^*phi hat ^ n$ is the closed form for sequence A_n.
Find c.
Note: phi is the golden ratio, phi hat is the complex conjugate

4. Let x, y, and z be positive real numbers that satisfy
2 log_x (2y) = 2 log_{2x} (4z) = log_{2x^4} (8yz) ≠ 0.
The value of xy^5z can be expressed in the form 1/(2^(p/q)), where p and q are relatively prime positive integers. Find p + q.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Geometric Series

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi aleph_zero;

There is a unique ordered pair (c,d) such that c*phi^n + d^*phi hat ^ n$ is the closed form for sequence A_n.

What does ^* mean?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

Hi aleph_zero;

There is a unique ordered pair (c,d) such that c*phi^n + d^*phi hat ^ n$ is the closed form for sequence A_n.

What does ^* mean?

Sorry, that should just be a * (multiplication)

@thickhead
I can't find x though...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

Let $$N = \sum_{k = 1}^{1000}k(\lceil \log_{\sqrt {2}}k\rceil - \lfloor \log_{\sqrt {2}}k \rfloor). $$
Find $N$.

How do I use LaTeX?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi;

How do I use LaTeX?

Put your Latex in between the math tags.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

6. Find all ordered pairs of real numbers

such that

.

How did you get the answer to the above?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

First, were all the other answers correct?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleph_zero

Member

Registered: 2016-08-11

Posts: 10

Re: Geometric Series

Yes, they were all correct. I understand how to get 3 and 5 now, but I still don't get 4. Thanks for all your help!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi;

Most of these contest problems are really hard for people who do math in the classical way. They are not so tough when you work on them using the principles of experimental math. 4 took a lot of trial and error, this is discouraged among students even today. But it is still a powerful weapon outside the classroom and in industry.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Geometric Series

6. Find all ordered pairs of real numbers

such that

.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geometric Series

If you can compute it, then you understand it. Can you compute it?

Table[FromContinuedFraction[ContinuedFraction[0.008012018027, k]], {k,2, 6}]

{1/124,1/125,5/624,11/1373,16/1997}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometric Series

Hi;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geometric Series

I wasn't able to find more than one ordered pair for number 6, x = -1 and  y = 0. What did you find ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Geometric Series

Last edited by thickhead (2016-08-16 17:55:38)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}