Can anyone me to generalize and prove this if it is valid?

  • Thread starter mahmudarif
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In summary, the given conversation discusses the expansion of two binomial expressions and their simplified forms. The first equation expands (x+y)^4 to x^4-(X^3)y+(X^2)(y^2)-x(y^3)+y^4= (x+y)^4-5xy(x+y)^2+5(xy)^2. The second equation expands (x+y)^6 to x^6-(x^5)y+(X^4)(y^2)-(X^3)(y^3)+(X^2)(y^4)-x(y^5)+y^6=(x+y)^6-7xy(x+y)^4 +14((xy)^2)(x+y)^2 - 7
  • #1
mahmudarif
4
0
X^4-(X^3)y+(X^2)(y^2)-x(y^3)+y^4= (x+y)^4-5xy(x+y)^2+5(xy)^2

x^6-(x^5)y+(X^4)(y^2)-(X^3)(y^3)+(X^2)(y^4)-x(y^5)+y^6=(x+y)^6-7xy(x+y)^4 +14((xy)^2)(x+y)^2 - 7(xy)^3

......
If n is an odd prime then prove,

x^n-1 - X^(n-2).y+...-x.y^(n-2)+y^(n-1) = (x+y)^n-1 - nxy(x+y)^(n-3) +...(-1)^((n-1)/2) . n .(xy) ^ ((n-1)/2)


Thank you very much in advance for your assistance.
 
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  • #2
mahmudarif said:
X^4-(X^3)y+(X^2)(y^2)-x(y^3)+y^4= (x+y)^4-5xy(x+y)^2+5(xy)^2

x^6-(x^5)y+(X^4)(y^2)-(X^3)(y^3)+(X^2)(y^4)-x(y^5)+y^6=(x+y)^6-7xy(x+y)^4 +14((xy)^2)(x+y)^2 - 7(xy)^3

......
If n is an odd prime then prove,

x^n-1 - X^(n-2).y+...-x.y^(n-2)+y^(n-1) = (x+y)^n-1 - nxy(x+y)^(n-3) +...(-1)^((n-1)/2) . n .(xy) ^ ((n-1)/2)


Thank you very much in advance for your assistance.



Your post is very difficult to read in ASCII...I even didn't try. My advice: learn how to post here in LaTeX or else

attach some document, preferably PDF, where mathematical stuff appears decently.

DonAntonio
 
  • #3
I think this has been posted before recently but, for the love of me, I can't find the original post.

And I believe the suggestion was to learn about the binomial theorem; then try to expand (x+y)^4 and (x+y)^6.
 
  • #4
that's simple re arrangement of binomial expansions. for general proof, try general form of binomial expansions. you will get it easily.
 
  • #5
Or else use the "sup" and "sub" tags to create superscripts and subscripts:

[noparse]xy xy[/noparse]

xy xy
 
  • #6
Got the point. Thanks every one...
 

1. Can you explain the process of generalization and proof?

Generalization is the process of applying a finding or conclusion from a specific case to a larger group or population. Proof, on the other hand, is the process of providing evidence or logical reasoning to support a claim or statement. In order to generalize and prove something, one must first gather data or information, analyze it, and then draw conclusions that can be applied to a larger context. This can involve statistical methods, logical arguments, or experiments.

2. How can I determine if something is valid for generalization and proof?

In order for something to be valid for generalization and proof, it must be supported by reliable and accurate data or evidence. This can include empirical data from experiments, statistical data from surveys or studies, or logical reasoning based on established principles. It is important to critically evaluate the quality of the data and evidence before making any conclusions or generalizations.

3. What are the potential limitations of generalization and proof?

There are several limitations to generalization and proof that should be considered. One limitation is the representativeness of the sample or data used for generalization. If the sample is not representative of the larger population, the conclusions may not be accurate. Another limitation is the potential for bias in the data or methods used. Additionally, certain phenomena may be too complex or multifaceted to be fully explained or proven through generalization and proof.

4. How can I improve the validity of my generalizations and proofs?

To improve the validity of generalizations and proofs, it is important to use a rigorous and systematic approach. This can include using appropriate methods for data collection and analysis, ensuring the sample is representative of the larger population, and considering potential biases in the data. It can also be helpful to consult with other experts in the field and to replicate the study or experiment to confirm the results.

5. Can generalization and proof be used in all scientific fields?

Generalization and proof can be used in many scientific fields, but they may not be applicable in all cases. Some fields, such as physics and chemistry, rely heavily on empirical data and mathematical proofs. Other fields, such as social sciences, may use a combination of data, logic, and theory to support generalizations and proofs. Ultimately, the use of generalization and proof depends on the nature of the research question and the available methods to address it.

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