Evaluate : `underset (r=1) overset (n) sum ""^nC_r 2^r`.

Prove that underset (r=0) overset (n) sum 3^r ""^nC_r =4^n .

Prove that underset (r=1) overset (5) sum "^5C_r=31 .

The value of ""^50C_4+ underset (r=1) overset (6) sum ""^(56-r)C_3 is :

The value of C (47, 4) + underset (r=1) overset (5) sum (52-r,3) is:

Equation x^(n)-1=0,ngt1,ninN, has roots 1,a_(1),a_(2),...,a_(n),. The value of underset(r=2)overset(n)sum(1)/(2-a_(r)), is

Equation x^(n)-1=0,ngt1,ninN, has roots 1,a_(1),a_(2),...,a_(n),. The value of underset(r=2)overset(n)sum(1)/(2-a_(r)), is

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

If R_(1) is the circumradius of the pedal triangle of a given triangle ABC, and R_(2) is the circumradius of the pedal triangle of the pedal triangle formed, and so on R_(3), R_(4) ..., then the value of underset( i=1)overset(oo)sum R_(i) , where R (circumradius) of DeltaABC is 5 is

The value of cot (underset(n=1)overset(23)sum cot^(-1) (1 + underset(k=1)overset(n)sum 2k)) is