Question 1012868
<pre>
It's easier to see when you write with larger parentheses enclosing
smaller parentheses like this:


{{{(x-(-y^"")^""^"")-(-(x^2-(-xy^"")-(-y^2)^"")^"")-(-(x^3-(-x^2y)^""-(-xy^2)-(-y^3))^"")-""*""*""*""}}}

First we remove the smallest parentheses: 

{{{(x+y^""^""^"")-(-(x^2+xy^""+y^2^"")^"")-(-(x^3+x^2y^""^""+xy^2+y^3)^"")-""*""*""*""}}}

Remove the medium-sized parentheses:

{{{(x+y^""^""^"")-(-x^2-xy^""-y^2^""^"")-(-x^3-x^2y^""^""-xy^2-y^3^""^"")-""*""*""*""}}}


Remove the largest parentheses and the remaining terms will also
have positive coefficients, so we put + before the three dots:

{{{x+y+x^2+xy+y^2+x^3+x^2y+xy^2+y^3+""*""*""*""}}}

Now that all terms have positive coefficients,
let's put parentheses back to separate the terms, and call it
expression (1)

(1)    {{{(x+y^"")+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+""*""*""*""}}}

Now we use the fact that {{{x^k-y^k}}} factors as 

{{{x^k-y^k}}}{{{""=""}}}{{{(x-y)(x^(k-1)+x^(k-2)y+x^(k-3)y^2+""*""*""*""+x^2y^(k-3)+xy^(k-2)+y^(k-1))}}}

Dividing both sides by (x-y)

{{{(x^k-y^k)/(x-y)}}}{{{""=""}}}{{{x^(k-1)+x^(k-2)y+x^(k-3)y^2+""*""*""*""+x^2y^(k-3)+xy^(k-2)+y^(k-1)}}}

Using that, expression (1) becomes:

{{{((x^2-y^2)/(x-y))+((x^3-y^3)/(x-y))+((x^4-y^4)/(x-y))+""*""*""*""}}}

And now we can conveniently write the final nth term. 
Notice that the exponent in the nth term will be n+1, because the
exponent in the 1st term is 2, the exponent in the 2nd term is 3,
etc.

{{{((x^2-y^2)/(x-y))+((x^3-y^3)/(x-y))+((x^4-y^4)/(x-y))+""*""*""*""+((x^(n+1)-y^(n+1))/(x-y))}}}


Factor out {{{1/(x-y)}}}

{{{(1/(x-y))((x^2-y^2)+(x^3-y^3)+(x^4-y^4)+""*""*""*""+(x^(n+1)-y^(n+1))^"")}}}

Remove the inner parentheses:

{{{(1/(x-y))(x^2-y^2+x^3-y^3+x^4-y^4+""*""*""*""+x^(n+1)-y^(n+1)^"")}}}

Rearranging the terms:

{{{(1/(x-y))(x^2+x^3+x^4+""*""*""*""+x^(n+1)-y^2-y^3-y^4-""*""*""*""-y^(n+1)^"")}}}

Put the x terms in parentheses and write the y terms in parentheses 
preceded by a - sign:

(2)    {{{(1/(x-y))((x^2+x^3+x^4+""*""*""*""+x^(n+1)^"")-(y^2+y^3+y^4+""*""*""*""+y^(n+1)^"")^"")}}}

Now we look at the sequences

{{{x^2+x^3+x^4+""*""*""*""+x^(n+1)}}} and {{{y^2+y^3+y^4+""*""*""*""+y^(n+1)}}}

These are geometric series with n terms, so we use the formula

{{{S[n]}}}{{{""=""}}}{{{a[n](r^n-1)/(r-1)}}}

with {{{x^2+x^3+x^4+""*""*""*""+x^(n+1)}}}, {{{r=x}}} and {{{a[1]=x^2}}}

{{{x^2+x^3+x^4+""*""*""*""+x^(n+1)}}}{{{""=""}}}{{{(x^2(x^n-1))/(x-1)}}}

Similarly,

{{{y^2+y^3+y^4+""*""*""*""+y^(n+1)}}}{{{""=""}}}{{{(y^2(y^n-1))/(y-1)}}}

Substituting those in expression (2),

{{{(1/(x-y))((x^2(x^n-1))/(x-1)-(y^2(y^n-1))/(y-1))^"")}}}

I'm not going to type out all the algebra, but this can be written

{{{(x^(n+2)*y-x^(n+2)-xy^(n+2)+y^(n+2)-x^2*y+x^2+xy^2-y^2)/((x-1) (y-1) (x-y))}}}

Edwin</pre>