Suppose n is a natural number such that `|i + 2i^2 + 3i^3 +...... + ni^n|=18sqrt2` where `i` is the square root of `-1`. Then n is

Prove that : Suppose that n is a natural number and I, F are respectively the integral part and fractional part of (7+4sqrt(3))^(n) . Then show that (i) I is an odd integer (ii) (I+F)(I-F)=1

Find sum of first n squared natural numbers i.e. 1^(2)+2^(2)+3^(2)+....+n^(2) .

Find sum of first n squared natural numbers i.e. 1^(2)+2^(2)+3^(2)+....+n^(2) .

For n being a natural number prove that 1.1!+2.2!+3.3!+.....+n.n! =(n+1)!-1 by applying P.M.I

If i= sqrt-1 and n is a positive integer , then i^(n) + i^(n + 1) + i^(n + 2) + i^(n + 3) is equal to

If a and b are positive integers such that N=(a+ib)^(3)-107i (where N is a natural number), then the value of a is equal to (where i^(2)=-1 )

If a and b are positive integers such that N=(a+ib)^(3)-107i (where N is a natural number), then the value of a is equal to (where i^(2)=-1 )

The least positive integer n such that ((2i)^n)/((1-i)^(n-2)),i=sqrt(-1) , is a positive integer,is ______