127.

Sum of the series n . 1 + (n - 1). 2 + (n - 2) . 3 + ........ + 1. n

The given series is: n. 1 + (n - 1) . 2 + (n - 2) . 3 + ............. + 1. n 

Let T_k denote its kth term

∴           T_k = [kth term of n, n - 1, .....]  [kth term of 1, 2, 3, ........]

                 = [n + (k - 1) (-1)] [1 + (k - 1)1] = (n - k + 1) (k) = nk - k² + k

              T_k = (n + 1)k - k²

Let   denote the sum of the series


∴          


                                        _{= 
   
                                         =}