Convert them to perfect squares

Algebra Level 3

( x + y ) + ( x 2 + x y + y 2 ) + ( x 3 + x 2 y + x y 2 + y 3 ) + ⋯ \large (x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+\cdots

Find the sum of the first n n terms of the summation above.

1 x + y [ x ( 1 − x n ) 1 − x ] − 1 x + y [ y ( 1 − y n ) 1 − y ] \dfrac1{x+y}[\dfrac{x(1-x^n)}{1-x}]-\dfrac1{x+y}[\dfrac{y(1-y^n)}{1-y}] 1 x + y [ x 2 ( 1 − x n ) 1 − x ] − 1 x + y [ y 2 ( 1 − y n ) 1 − y ] \dfrac1{x+y}[\dfrac{x^2(1-x^n)}{1-x}]-\dfrac1{x+y}[\dfrac{y^2(1-y^n)}{1-y}] 1 x − y [ x ( 1 − x n ) 1 − x ] − 1 x − y [ y ( 1 − y n ) 1 − y ] \dfrac1{x-y}[\dfrac{x(1-x^n)}{1-x}]-\dfrac1{x-y}[\dfrac{y(1-y^n)}{1-y}] 1 x − y [ x 2 ( 1 − x n ) 1 − x ] − 1 x − y [ y 2 ( 1 − y n ) 1 − y ] \dfrac1{x-y}[\dfrac{x^2(1-x^n)}{1-x}]-\dfrac1{x-y}[\dfrac{y^2(1-y^n)}{1-y}]

Sabhrant Sachan by Sabhrant Sachan , 21, India

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