Maths-
General
Easy

Question

If x equals sum subscript n equals 0 end subscript superscript infinity end superscript   a to the power of n end exponent comma y equals sum subscript n equals 0 end subscript superscript infinity end superscript   b to the power of n end exponent comma z equals sum subscript n equals 0 end subscript superscript infinity end superscript   left parenthesis a b right parenthesis to the power of n end exponent where a<1, b<1 then

  1. xy+xz=yz+x    
  2. xyz=x+y+z    
  3. xy+yz=xz+y    
  4. yz+zx=xy+z    

Hint:

We can write x as
x space equals space fraction numerator 1 over denominator 1 minus a end fraction space
rightwards double arrow a space equals space fraction numerator x minus 1 over denominator x end fraction
Similarly, we can write y as
y space equals space fraction numerator 1 over denominator 1 minus b end fraction space
rightwards double arrow b space equals space fraction numerator y minus 1 over denominator y end fraction

Similarly, we can write z as
z space equals space fraction numerator 1 over denominator 1 minus a b end fraction space
rightwards double arrow a b space equals space fraction numerator z minus 1 over denominator z end fraction

Multiply ab and equate the values and simplify

The correct answer is: yz+zx=xy+z


    Given : x equals sum subscript n equals 0 end subscript superscript infinity end superscript   a to the power of n end exponent comma y equals sum subscript n equals 0 end subscript superscript infinity end superscript   b to the power of n end exponent comma z equals sum subscript n equals 0 end subscript superscript infinity end superscript   left parenthesis a b right parenthesis to the power of n end exponent , where a<1, b<1 
    To find : A relationship between x, y  and z
    We can write x as
    x space equals space fraction numerator 1 over denominator 1 minus a end fraction space
rightwards double arrow a space equals space fraction numerator x minus 1 over denominator x end fraction
    Similarly, we can write y as
    y space equals space fraction numerator 1 over denominator 1 minus b end fraction space
rightwards double arrow b space equals space fraction numerator y minus 1 over denominator y end fraction

    Similarly, we can write z as
    z space equals space fraction numerator 1 over denominator 1 minus a b end fraction space
rightwards double arrow a b space equals space fraction numerator z minus 1 over denominator z end fraction

    a space cross times space b space equals space open parentheses fraction numerator x minus 1 over denominator x end fraction close parentheses cross times space open parentheses fraction numerator y minus 1 over denominator y end fraction close parentheses space equals space open parentheses fraction numerator z minus 1 over denominator z end fraction close parentheses
    Solving the above, we get
    x y z space minus space x y space equals space left parenthesis x z minus z right parenthesis left parenthesis y minus 1 right parenthesis

x y z space minus space space x y space equals space x y z space minus x z space minus z y space plus space z

F u r t h e r space s i m p l i f y i n g comma space w e space g e t

left parenthesis x y space plus space z right parenthesis space equals left parenthesis z x space plus y z right parenthesis

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