Let x1x x2x x3,........ Are In HP then value of summation k=1to k=20 (-1)^k.(xk+xk+1/xk-xk+1) equals to

Hi , thank you for asking the question      
consider  (-1)_^k(x_^k+x_^k+1/x_^k  - x ^_k+1)     ~1
now ,as they are in H.P their inverse is in A.P and let common difference is r

1/x_^k+1=1/x_^k+r

1/x_^k+1 – 1/x_^k =r

1/( x_^k    –  x_^k+1  )=1/(r* x^_k  *x_^k+1)   ~2

substitute 2 in 1 then we get

(-1)^k(x_^k+x_^k+1/r* x_^k  *x_^k+1)=  (-1)^k(1/r)(1/ x_^k  +1/ x_^k+1 )

so the sequence we come as    -1(1/r)(1/x₁+1/x₂)+1/r(1/x₂+1/x₃)............
(1/r)[-1/x₁+1/x₂₁]=     (1/r)[1/x₂₁-1/x₁]

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