Proving Homework Statement: Expanding Power of 2 over 3

  • Thread starter eddybob123
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In summary, the conversation is about finding a solution to a math competition problem involving representing positive integers in a specific form. The method suggested involves writing the number in base three and subtracting it from a simpler term, but the complexity may still arise due to the use of powers.
  • #1
eddybob123
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Homework Statement


In the link

Homework Equations


In the link


The Attempt at a Solution


First I tried expanding the sum as a power of 2 over a power of 3, but I failed.
 

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  • #2
I was not able to open the file. Why not just put the problem and your work directly into the form?
 
  • #3
Mark44 said:
I was not able to open the file. Why not just put the problem and your work directly into the form?

The statement was: prove that every positive integer can be represented in the form
[tex] \frac{2^a}{3^b} -\sum_{k=0}^b \frac{2^{c_k}}{3^k}, \\
\text{where } a, b, c_k \text{ are integers },1 = c_0 < c_1 < c_2 < \cdots . [/tex]

It does not say whether or not the a and b are allowed to vary with the integer n to be represented, or whether a and/or b are supposed to be fixed.

RGV
 
Last edited:
  • #4
Start by writing the number in base three. Then you have the number as a sum of negative powers of three with all numerators either 1 or 2. Those with are what you want. If there is a numerator of 1, combine fractions.
 
  • #5
@Ray a and b are allowed to vary, but they have to be integral.
@Mark I do not know how to use the codes
@HallsofIvy I am pretty sure that the answer does not involve different bases. The expression is merely a subtraction of powers of 2 over powers of 3.
 
  • #6
Truth be told, this isn't a homework question for school. This is merely a question on a recent math competition that I have not answered and I want to know.
 
  • #7
eddybob123 said:
@Ray a and b are allowed to vary, but they have to be integral.
@Mark I do not know how to use the codes
@HallsofIvy I am pretty sure that the answer does not involve different bases. The expression is merely a subtraction of powers of 2 over powers of 3.
Which is a question of base 3, pretty much by definition!
 
  • #8
However it will get just as complicated as in base ten because of all the powers of 2
 
  • #9
Suppose we can approach it in a different way. We can just sum up the terms in the summation and define a "simpler" term, which we can then subtract from the first term.
 
  • #10
eddybob123 said:
However it will get just as complicated as in base ten because of all the powers of 2

No, it won't. Every integer, written is base three, is, by definition, of the form [itex]\sum_{i=0}^N a_i/3^i[/itex] where each [itex]a_i[/itex], because it is a base 3 digit, is either 0, or 1= 20, or 2= 21.
 
  • #11
I am just looking for an answer, nothing more.
 

What does "Expanding Power of 2 over 3" mean?

The phrase "Expanding Power of 2 over 3" refers to the mathematical process of raising the number 2 to increasingly higher powers, and then dividing each result by 3. This process can be used to find the sum of an infinite geometric series with a starting term of 2 and a common ratio of 1/3.

Why is this process important?

This process is important because it allows us to find the sum of an infinite geometric series, which can be applied to various real-world problems such as calculating compound interest, population growth, and radioactive decay.

How do I prove this homework statement?

To prove this homework statement, you will need to use mathematical induction. This involves showing that the statement is true for a specific starting point (usually n = 1), and then proving that if it is true for any value k, it is also true for the next value (k+1). This process must be repeated infinitely to prove the statement for all possible values of n.

What are some tips for solving this type of problem?

One tip for solving this type of problem is to start by writing out the first few terms of the series and looking for patterns. This can help you come up with a general formula for the sum. Additionally, it is important to carefully follow the steps of mathematical induction and clearly explain each step in your proof.

Can this process be applied to other numbers besides 2 and 3?

Yes, this process can be applied to any number raised to a power and then divided by a different number. The key is to find the common ratio between each term in the series and use that to create a formula for the sum. However, the specific values used in the homework statement may change depending on the numbers involved.

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