Find the value of the following
`i^(n)+ i^(n +1) + i^(n+2) + i^(n+3)`, for all `n in N`

Find the value of n, if 12 xx ^((n-1))P_3= 5 xx ^((n+1))P_3

Complete the following (i) 'Li+N_2 gives

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

The value of Sigma_(k=0)^(n)(i^(k)+i^(k+1)) , where i^(2)=-1 , is equal to

Find the sum of the series 1.n + 2 .(n-1) + 3.(n-2) + cdots + ( n-1) .2 + n.1.

(a)Prove the following by using the principle of mathematical induction for all n in N:2^n<3^n .

(a)Prove the following by using the principle of mathematical induction for all n in N:2^1+2^2+……..+2^n=2^(n+1)-2 .

Let f: N to N be defined by f(n) = { (((n+1)/2)if n is odd), (n/2 if n is even) :} for all n in N State whether the function f is bijective. Justify yur answer.

Matrix A such that A^(2) = 2A - I , where I is the identity matrix, then for n ge 2, A^(n) is equal to a) 2^(n - 1) A - (n - 1) I b) 2^(n - 1)A - I c) n A - (n - 1) I d) nA - I

Using principle of mathematical induction prove that n(n+1)(n+5) is a multiple of 3 for all n in N .