The sequence N of natural numbers is divided into classes as follows :
Show that the sum of the numbers in the nth row is `n(2n^2+1)` .

The sequence of odd natural numbers is divided into groups 1,3,5,7,9,11,"…." and so on. Show that the sum of the numbers in nth group is n^(3) .

If the sum of n natural numbers is one sixth of their squares, then n is :

Find the sum to n terms of the series whose nth term is n^(2) + 2^(n) .

Show that the sequence whose n^(th) term is 2n^2 + n + 1 is not an A.P.

On the set N of all natural numbers, a relation R is defined as follows: AA n,m in N , n R m Each of the natural numbers n and m leaves the remainder less than 5.Show that R is an equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

Find the sum of first n natural numbers.

Find the sum of n terms of the series whose nth term is : 2^(n-1)-1 .

Find the sum of n terms of the series whose nth terms is (i) n(n-1)(n+1) .

The natural numbers arearranged innthe form given below The rth group containing 2^(r-1) numbers. Prove that sum of the numbers in the nth group is 2^(n-2)[2^(n)+2^(n-1)-1] .

Find the sum of n terms of the series whose nth term is : 2^(n+1)+4(n+1)(n-2) .