Prove that
`3.C_(0)^(2)+7.C_(1)^(2)+11.C_(2)^(2)+…..+(4n+3).C_(n)^(2)=(2n+3).""^(2n)C_(n).`

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that Prove that : C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+….+C_(n)^(2)=""^(2n)C_(n)

Prove that C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+….+(-1)^(n).C_(n)^(2)={{:((-1)^(n//2)""^(n)C_(n//2)",","if n is even"),(" "0" ,","if n is odd"):}

Prove that (C_(0) +C_(1)+C_(2)+….+C_(n))^(2)=1 +""^(2n)C_(1) +""^(2n)C_(2) +…..+""^(2n)C_(2n)

Prove that : Prove that C_(0)+3.C_(1)+3_(2).C_(2)+……+3^(n).C_(n)=4^(n)

Prove that following C_(0)+2.C_(1)+4.C_(2)+8.C_(3)+…+2^(n).C_(n)=3^(n)

Prove that C _(0) +2 . C _(1) + 2 ^(2) . C _(2) + .......+ 2 ^(n) . C _(n) = 3 ^(n).

Prove that 5.C_(0) +5^(2).(C_(1))/(2) +5^(3).(C_(2))/(3)+…+5^(n+1).(C_(n))/(n+1)=(6^(n+1)-1)/(n+1) Hence show that 5.C_(0) +(5^(2))/(2).C_(1)+(5^(3))/(3).C_(2)+….+(5^(11))/(11).C_(10)=(6^(11)-1)/(11)

Prove that C_(0)+3.C_(1)+5.C_(2)+….+(2n+1).C_(n)=(2n+2).2^(n-1).

Prove the C_(0)+2.C_(1)+4.C_(2)+8.C_(3)+…+2^(n).C_(n)=3^(n)