If `Delta_(r)=|{:(r,r-1),(r-1,r):}|` where is a natural number, the value of `root(10)(sum_(r=1)^(1024))Delta_(r)` is

The value of sum_(r=-3)^(1003)i^(r)(where i=sqrt(-1)) is

sum_(r=1)^n(2r+1)=...... .

Statement -1 If Delta(r)=|{:(r,r+1),(r+3,r+4):}| then sum_(r=1)^(n) Delta(r)=-3n Satement-2 If Delta(r)=|{:(f_(1)(r),f_(2)(r)),(f_(3)(r),f_(4)(r)):}| Sigma_(r=1)^(n) Delta (r)={:abs((Sigma_(r=1)^(n)f_1(r),Sigma_(r=1)^(n)f_2(r)),(Sigma_(r=1)^(n)f_3(r),Sigma_(r=1)^(n) f_4(r))):}

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

Evaluate sum_(r=1)^(n)rxxr!

The value of the lim_(n->oo)tan{sum_(r=1)^ntan^(- 1)(1/(2r^2))} is equal to

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

If alpha_(1), alpha_(2), alpha_(3), beta_(1), beta_(2), beta_(3) are the values of n for which sum_(r=0)^(n-1)x^(2r) is divisible by sum_(r=0)^(n-1)x^(r ) , then the triangle having vertices (alpha_(1), beta_(1)),(alpha_(2),beta_(2)) and (alpha_(3), beta_(3)) cannot be

If t_(1)=1,t_(r )-t_( r-1)=2^(r-1),r ge 2 , find sum_(r=1)^(n)t_(r ) .

Let S_(n) denote the sum of the cubes of the first n natural numbers and S'_(n) denote the sum of the first n natural numbers, then underset(r=1)overset(n)Sigma ((S_(r))/(S'_(r))) equals to