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Prove that (C0+C1)(C1+C2)(C2+C3)(C3+C...

Prove that `(C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n)` = `(C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)`

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`LHS = (C_(0) + C_(1)) (C_(1)+ C_(2)) (C_(2) + C_(3))...(C_(n-1) + C_(n))`
`= prod_(r=1)^(n) (C_(r-1) + C_(r)) = prod_(r=1)^(n) (""^(r+1)C ) " " [ because ""^(n)C_(r) + ""^(n)C_(r-1) = ""^(n+1)C_(r)]`
`= prod_(r=1)^(n) ((n+1)/(r))""^(n)C_(r-1) " "[because ""^(n)C_(r) = (n)/(r) ""^(n-1)C_(r-1)]`
`= prod_(r=1)^(n)(n+1) . = prod_(r=1)^(n)(1)/(r).prod_(r=1)^(n)C_(r-1)`
`= (n+1)^(n) . (1)/(n!) (C_(0)C_(1) C_(2) ...C_(n-1))`
` = ((n+1)^(n))/(n!) (C_(0)C_(1) C_(2) ...C_(n-1))` = RHS
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