If n is a positive integer, prove that `underset(nrarrinfty)Lt(Sigman^2)/(n^3)=1/3`

If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is

For every positive integer n, prove that 7^n-3^n is divisible by 4.

For an positive integer n, prove that : i^(n) + i^(n+1) + i^(n+2) + i^(n+3) + i^(n+4) + i^(n + 5) + i^(n+6) + i^(n+7) = 0 .

If n is a positive integer and (3sqrt(3)+5)^(2n+1)=l+f where l is an integer annd 0 lt f lt 1 , then

If n is a positive integer satisfying the equation 2+(6*2^(2)-4*2)+(6*3^(2)-4*3)+"......."+(6*n^(2)-4*n)=140 then the value of n is

underset(nrarrinfty)lim (1^2+2^2+3^2+………+n^2)/n^3 is equal to:

Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

If n is any positive integer , show that 2^(3n +3) -7n - 8 is divisible by 49 .

If a and b are distinct integers, prove that a - b is a factor of a^(n) - b^(n) ,whenever n is a positive integer.