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Prove that : sum(i=0)^r((n+i),(k))=((n+r...

Prove that : `sum_(i=0)^r((n+i),(k))=((n+r+1),(k+1))-((n),(k+1))`

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Prove that sum_(k=1)^n 1/(k(k+1))=1−1/(n+1) .

If ((n),(r )) "denotes " ""^nC_r then (a) Evalutae : 2^(15)((30),(0))((30),(1))-2^(14)((30),(1))((29),(14))+2^(13)((30),(2))((28),(13)).......-((30),(15))((15),(0)) ( b) Prove that : Sigma_(r=1)^(n) ((n-1),(n-r))((n),(r))=((2n-1),(n-1)) ( c) Prove that : ((n),(r))((r),(k))=((n),(k))((n-r),(r-k))

Prove that sum_(k=1)^(n)(1)/(k(k+1))=1-(1)/(n+1).

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Prove that sum_(k=1)^(n-r ) ""^(n-k)C_(r )= ""^(n)C_( r+1) .

prove that sum_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))