Show that `1!+2!+3!++n !` cannot be a perfect square for any `n in N ,ngeq4.`

Show that 3^(n) xx 4^(m) cannot end with the digit 0 or 5 for any natural numbers ‘n’and 'm'

Matrix A such that A^2=2A-I ,w h e r eI is the identity matrix, the for ngeq2. A^n is equal to 2^(n-1)A-(n-1)l b. 2^(n-1)A-I c. n A-(n-1)l d. n A-I

Show that the square of any positive integer of the form 3m or 3m + 1 for some integer n.

Prove that by using the principle of mathematical induction for all n in N : 1.3+3.5+5.7+....+(2n-1)(2n+1)= (n(4n^(2)+6n-1))/(3)

Prove that by using the principle of mathematical induction for all n in N : 1.3+ 2.3^(2)+ 3.3.^(3)+ ....+ n.3^(n)= ((2n-1)3^(n+1)+3)/(4)

Show that one and only one out of n, n + 2 or n + 4 is divisible by 3, where n is any positive integer

Prove that by using the principle of mathematical induction for all n in N : (1)/(1.4)+ (1)/(4.7)+(1)/(7.10)+...+ (1)/((3n-2)(3n+1))= (n)/(3n+1)

Show that 5^(2n) -1 is divisible for all n in N by 24.

If (1)/(x^(4))-(6)/(x^(3))+(13)/(x^(2))+(m)/(x)+ n is a perfect square, find the values of m and n.

Show that 10^(2n)-1 is divisible by 11 for all n in N .