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

mahmudarif

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.

DonAntonio

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

dodo

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.

Akshay_Anti

that's simple re arrangement of binomial expansions. for general proof, try general form of binomial expansions. you will get it easily.

lpetrich

Or else use the "sup" and "sub" tags to create superscripts and subscripts:

x[sup]y[/sup] x[sub]y[/sub]

x^y x_y

mahmudarif

Got the point. Thanks every one...