Question

find the sum of the following series (x+y)+(x^2+y^2+xy)+(x^3+x^2y+xy^2+y^3)+.........to n terms​

Answer 1

Answer:

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Answer 2

Answer:

(x²(xⁿ⁺²-1)/(x²-1) -y²(yⁿ⁺²-1)/(y²-1))/(x-y)

Step-by-step explanation:

Given:

(x+y)+(x^2+y^2+xy)+(x^3+x^2y+xy^2+y^3)+.........to n

If we multiply the series by (x-y) it changes to:

(x²-y²)+(x³-y³)+...+(xⁿ⁺¹-yⁿ⁺¹)

And this is easier to calculate as a sum of :

x²+x³+...+xⁿ⁺¹ and  -(y²+y³+...+yⁿ⁺¹)

So, we have:

x²+x³+...+xⁿ⁺¹ = x²(xⁿ⁺²-1)/(x²-1) and

-(y²+y³+...+yⁿ⁺¹)= -y²(yⁿ⁺²-1)/(y²-1)

Added up:

(x²-y²)+(x³-y³)+...+(xⁿ⁺¹-yⁿ⁺¹)= x²(xⁿ⁺²-1)/(x²-1) -y²(yⁿ⁺²-1)/(y²-1)

And we need to divide it by (x-y) we used in the beginning:

(x+y)+(x^2+y^2+xy)+(x^3+x^2y+xy^2+y^3)+......=

(x²(xⁿ⁺²-1)/(x²-1) -y²(yⁿ⁺²-1)/(y²-1))/(x-y)