Home
Class 11
MATHS
Let x=1+a+a^2+...and y=1+b+b^2+..., wher...

Let `x=1+a+a^2+...`and `y=1+b+b^2+...`, where `|a|<1`and `|b|<1`. Prove that `1+a b+a^2b^2+...=(x y)/(x+y-1)`

Text Solution

Verified by Experts

`x=1+a+a^2+...`
`x = 1/(1-a)`
`1/x = (1-a)`
`a=1-1/x`
`a=(x-1)/x`
and `y=1+b+b^2+...`
`y = 1/(1-b)`
`1/y = (1-b)`
...
Promotional Banner

Topper's Solved these Questions

  • SEQUENCES AND SERIES

    NCERT|Exercise EXERCISE 9.1|14 Videos

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise EXERCISE 2.3|5 Videos

  • SETS

    NCERT|Exercise EXERCISE 1.5|7 Videos

Similar Questions

Explore conceptually related problems

Let x = [(a + 2b)/(a+b)] " and " y = a/b , where a and b are positive integers . If y^2 gt 2 , then

If =1+a+a^(2)+oo, where |a|< 1andy =1+b+b^(2)+oo, where |b|<1 prove that: 1+ab+a^(2)b^(2)+oo=(xy)/(x+y-1)

If (x)/(a)-(y)/(b) tan theta =1 and (x)/(a) tan theta +(y)/(b)=1 , then the value of (x^(2))/(a^(2))+(y^(2))/(b^(2)) is

Let x , y be two variables and x >0, x y=1 , then minimum value of x+y is (a) 1 (b) 2 (c) 2 1/2 (d) 3 1/3

Let f(x)=(1+b^(2))x^(2)+2bx+1 and let m(b) be the minimum value of f(x). As b varies,the range of m(b) is [0,} b.(0,(1)/(2)) c.(1)/(2),1 d.(0,1]

Find the area bounded by the ellipse (x ^(2))/( a ^(2)) + ( y ^(2))/( b ^(2)) =1 and the ordinates x = ae and x =0, where b ^(2) =a ^(2) (1-e ^(2)) and e lt 1.

Let x_(1),x_(2) are the roots of the quadratic equation x^(2) + ax + b=0 , where a,b, are complex numbers and y_(1), y_(2) are the roots of the quadratic equation y^(2) + |a|yy+ |b| = 0 . If |x_(1)| = |x_(2)|=1 , then prove that |y_(1)| = |y_(2)| =1