सिद्ध कीजिए कि :
`(a) (1+i)^(5)(1+(1)/(i))^(5)=32` `(b) (1-i)^(n)(1-(1)/(i))^(n)=2^(n)`

Simplify the following : (i) 1+ i^(5)+i^(10)+i^(15) (ii) (1+i)^(4)+(1+(1)/(i))^(4) (iii) i^(n)+i^(n+1)+i^(n+2)+i^(n+3)

Simplify the following : (i) 1+ i^(5)+i^(10)+i^(15) (ii) (1+i)^(4)+(1+(1)/(i))^(4) (iii) i^(n)+i^(n+1)+i^(n+2)+i^(n+3)

Show that (1-i)^(n)(1-(1)/(i))^(n)=2^(n) for all n in N

(1+i)^(2 n)+(1-i)^(2 n), n in z is

Find the integral values of n for the equations : (a) (1+i)^(n)=(1-i)^(n) (b) (1-i)^(n)=2^(n)

If I(m,n)=int_0^1 t^m(1+t)^n.dt , then the expression for I(m,n) in terms of I(m+1,n-1) is: (a) (2^(n))/(m+1)-n/(m+1)I(m+1,n-1) (b) n/(m+1)I(m+1,n-1) (c) (2^(n))/(m+1)+n/(m+1)I(m+1,n-1) (d) m/(m+1)I(m+1,n-1)

The value of 1/(i^n) + 1/(i^(n + 3)) + 1/(i^(n + 2)) + 1/(i^(n + 1)) is :

For positive integer n_1,n_2 the value of the expression (1+i)^(n1) +(1+i^3)^(n1) (1+i^5)^(n2) (1+i^7)^(n_20), where i=sqrt-1, is a real number if and only if (a) n_1=n_2+1 (b) n_1=n_2-1 (c) n_1=n_2 (d) n_1 > 0, n_2 > 0

For positive integer n_1,n_2 the value of the expression (1+i)^(n1) +(1+i^3)^(n1) (1+i^5)^(n2) (1+i^7)^(n_20), where i=sqrt-1, is a real number if and only if (a) n_1=n_2+1 (b) n_1=n_2-1 (c) n_1=n_2 (d) n_1 > 0, n_2 > 0

For positive integers n_(1), n_(2) the value of the expression (1+i)^(n_(1)) + (1+ i^(3))^(n_(1)) + (1+ i^(5))^(n_(2)) + (1+ i^(7))^(n_(2)) , where i= sqrt-1 is a real number if and only if