Given series is:

1+ (1+2) + (1+2+3) + ................

The n^th term of the series is

      a_n = 1+2+3+.........n

=> a_n = n(n+1)/2

=> a_n = (n² + n)/2

Now, S_n = ∑ a_k , where 1<= k<= n 

=>   S_n = ∑(n² + n)/2

=>   S_n = (1/2)*∑(n² + n)

=>  S_n = (1/2)*{∑n² + ∑n}

=>  S_n = (1/2)*[{n*(n+1)*(2n+1)}/6 + n*(n+1)/2}

=>  S_n = (1/2)*n*(n+1)*[(2n+1)/6 + 1/2]

=>  S_n = (1/2)*n*(n+1)*[{(2n+1) + 3}/6]

=>  S_n = (1/2)*n*(n+1)*{(2n+4)/6}

=>  S_n = (1/12)*n*(n+1)*(2n+4)

So, the sum of the given series is (1/12)*n*(n+1)*(2n+4)