Question

if x=a+a/r+a/r^2+.... , y= b-b/r+b^2/r^2.... , z= c+c/r^2+c/r^4+...., then prove that xy/z=ab/c

Answer 1

Given :  x=a+a/r+a/r^2+......to infinity. y=b-b/r+b/r^2+.......to infinty , z=c+c/r^2+c/r^4+.....to infinty

To find : prove that xy/z=ab/c​

Solution:

x  = a + a/r  +  a/r² + .....................+ ∞

First term = a  , Common Ratio = 1/r

=> x =  a/(1 - 1/r)  =  ra/(r - 1)

y  = b - b/r  + b/r² + .....................+ ∞

First term = b  , Common Ratio = -1/r

=> y =  b/(1 - (-1/r))  =  rb/(r + 1)

z=c+c/r²+c/r⁴+..........................+ ∞

First term = c  , Common Ratio = 1/r²

z = c/(1 - (1/r²) = r²c/(r² - 1)

LHS = xy/z    

=   (ra/(r - 1) )( rb/(r + 1)) / ( r²c/(r² - 1))

= r²ab(r² - 1)/ r²c/(r² - 1)

= ab/c

= RHS

QED

Hence Proved

xy/z=ab/c​

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Answer 2

Step-by-step explanation:

See your proof in the attachment.