Question

Let x= 1+a+a^2+.......And y=1+b+b^2 prove that 1+ab+a^2b^2=xy/x+y-1

Answer 1

Answer:

proved

Step-by-step explanation:

Given Let x= 1+a+a^2+.......And y=1+b+b^2 prove that 1+ab+a^2b^2=xy/x+y-1

Answer 2

x=1+a+a2+a3+⋯

y=1+b+b2+b3+⋯

Letz=1+ab+a2b2+a3b3+⋯

∵a and b are proper fractions

∴a<1andb<1

x=1+a+a2+a3+⋯is the sum of an infinite geometric progression

with first term 1 and common ratioa<1

∴sum x=a1−r=11−a

y=1+b+b2+b3+⋯is the sum of an infinite geometric progression

with first term 1 and common ratiob<1

∴sum y=a1−r=11−b

z=1+ab+a2b2+a3b3+⋯is the sum of an infinite geometric progression

with first term 1 and common ratioab<1

∴sum z=a1−r=11−ab(1)

x=11−a⟹1x=1−a⟹1x−1=−a(2)

y=11−b⟹1y=1−b⟹1y−1=−b

⟹−1y+1=b(3)

by (2) x (3)

(1x−1)(−1y+1)=−ab

⟹(1−xx)(y−1y)=−ab

⟹((1−x)(y−1)xy)=−ab

⟹(y−1−xy+xxy)=−ab

⟹1+(y−1−xy+xxy)=1−ab

⟹(xy+y−1−xy+xxy)=1−ab

⟹(x+y−1xy)=1−ab

⟹xyx+y−1=11−ab=zby (1)

⟹1+ab+a2b2+a3b3+⋯=xyx+y−1