Write $x^{n-1} + x^{n-2}y + \cdots + xy^{n-2} + y^{n-1}$ using summation notation

Question

How does one write $x^{n-1} + x^{n-2}y + \cdots + xy^{n-2} + y^{n-1}$ using summation notation $\sum$? Use two indices ( and some sort of magic to indicate that the outer index should decrement)?

$$\sum_{\substack{i=0 \\ j=n-1}}^{\substack{0 \\ n-1}} x^j y^i$$

Is it possible to write it using only one index? Should it be written with only one index?

Answer 1

$$ \sum_{i=0}^{n-1} x^{n -1-i} y^i $$ But if you really want to use more than one index, you could try this: $$ \sum_{\begin{smallmatrix} i,j\,\in\,\{0,\,\ldots\,,\,n-1\} \\ i+j = n - 1 \end{smallmatrix}} x^i y^j $$

Answer 2

It may help to note what is implicit here; namely $$ S=x^{n-1} + x^{n-2}y + \cdots + xy^{n-2} + y^{n-1}=x^{n-1}y^0+x^{n-2}y^1+\cdots+x^1y^{n-2}+x^0y^{n-1}. $$ Then it is rather straightforward: $$ S=\sum_{i=0}^{n-1}x^{n-1-i}y^i. $$

Answer 3

Even though it's solved already I'll just leave here this $\dfrac{x^n-y^n}{x-y}$