If `n_(j_(r))=((1-xn)(1-x(n-1))(1-x^(n-2)).......(1-x^(n-r+1)))/((1-x)(1-x^2)(1-x^3)........(1-x^r)),` prove that `n_(j_(n-r) = n_(j_(r.))`

If x+y=1, prove that sum_(r=0)^n .^nC_r x^r y^(n-r) = 1 .

If x+y=1, prove that sum_(r=0)^n .^nC_r x^r y^(n-r) = 1 .

(1+x)(sum_(r=0)^(n)nC_(r)x^(r))+(1+x^(2))(sum_(r=0)^(n-1)(n-1)C_(r)x^(r))+...(1+x^(n))(sum_(r=0)^(1)1C_(r)x^(r))=a_(0)+a_(1)x+, then the value of (a_(1))/(a_(n)+1)+(a_(1))/(a_(n))+(a_(2))/(a_(n-1))+......+(a_(n+1))/(a_(0)) is equal to

if ((n),(r)) + ((n),(r-1)) = ((n + 1),(x)), then x = ....

sum_(r=1)^(n)tan((x)/(2^(r)))sec((x)/(2^(r-1)));r,n in N

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), then prove that C_(1)+2c_(2)+3C_(1)+...+nC_(n)=n2^(n-1)...

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(1)+2C_(2)+3C_(3)+....+nC_(n)=n2^(n-1)

The coefficient of x^r [0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+.... +(x+2)^(n-1) is a.^n C_r(3^r-2^n) b.^n C_r(3^(n-r)-2^(n-r)) c. ^n C_r(3^r+2^(n-r)) d. none of these

If x_1, x_2, x_3,...... x _(2n) are in A.P , then sum _(r=1)^(2n) (-1)^(r+1) x_r^2 is equal to (a) (n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (b) (2n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (c) (n)/((n-1))(x _(1)^(2) -x _(2n) ^(2)) (d) (n)/((2n+1))(x _(1)^(2) -x _(2n) ^(2))