if sn =(x+y)+(x^2+xy+y^2)+(x^3+x^2y +xy^2+y^3)+...n terms then prove that (x-y) =[x^2(x^n-1)/x-1 - y^2(y^n-1)/y-1
Answers
Given : sₙ =(x+y)+(x²+xy+y²)+(x³+x²y +xy²+y³)+...n terms
To Prove : (x - y) = [x²(xⁿ-1)/(x-1) - y²(yⁿ-1)/(y-1)]/sₙ
Solution:
sₙ =(x+y)+(x²+xy+y²)+(x³+x²y +xy²+y³) +.............+ n terms
Multiplying both sides by x - y
=> (x - y) sₙ = (x - y) [(x+y)+(x²+xy+y²)+(x³+x²y +xy²+y³) +.............+ n terms ]
=> (x - y) sₙ = (x² - y ²)+ (x³ - y³) + (x⁴ - y⁴) +.............+ (n terms )
=> (x - y) sₙ = (x² + x³ + x⁴ + .....+ n terms) - (y² + y³ + y⁴ + .....+ n terms)
using gp summation formula
a(rⁿ - 1)/(r - 1)
a = x² r = x & a = y² , r = y
=> (x - y) sₙ = x²(xⁿ - 1)/(x - 1) - ( y²(yⁿ - 1)/(y - 1))
=> (x - y) = [x²(xⁿ - 1)/(x - 1) - ( y²(yⁿ - 1)/(y - 1))]/sₙ
(x - y) = [x²(xⁿ - 1)/(x - 1) - ( y²(yⁿ - 1)/(y - 1))]/sₙ
QED
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