Question

sum the series

(x+y) + (x2 + xy + y2 ) + (x3 + x2y + xy2 + y3) +..........10 nth term

Answer 1

HELLO DEAR,

given series are:(x + y) + (x² + x²y + y²) + (x³ + x²y + xy² + y³)+........10th term.

now, we have to find Sum of the series,
let us multiply and divide by (x - y) this will make easy to Solve,
We get, s₁₀ = 1/(x - y) * (x - y)(x + y) + (x - y)(x² + x²y + y²) + (x - y)(x³ + x²y + xy² + y³) +...........10terms.
We can see this is in the form of (xⁿ - yⁿ)

now, s₁₀ = 1/(x - y) * (x² - y²) + (x³ - y³) + (x⁴ - y⁴) +...........(x¹⁰ - y¹⁰).

now, let us separate the series,we get


(x - y)s₁₀ = (x² + x³ + x⁴+........+x¹⁰terms) - (y² + y³ + y⁴ +.........y¹⁰terms)​

(x - y)s₁₀ = [x²(1 - x¹⁰)/(1 - x)] - [y²(1 - y¹⁰)/(1 - y)]

s₁₀ = 1/(x - y) [x²(1 - x¹⁰)/(1 - x) - y²(1 - y¹⁰)/(1 - y)]

HENCE, the sum of series is: s₁₀ = 1/(x - y) [x²(1 - x¹⁰)/(1 - x) - y²(1 - y¹⁰)/(1 - y)]

I HOPE ITS HELP YOU DEAR,
THANKS