Step-by-step Solution:

\[ \text{Required mean} = \frac{\sum (1+i)x_i}{n} = \frac{\sum (1+i)i}{n} = \frac{\sum (i^2 + i)}{n} = \frac{\sum i^2 + \sum i}{n} \] \[ = \frac{\frac{1}{6} n (n+1) (2n+1) + \frac{1}{2} n (n+1)}{n} = \frac{(n+1)(n+2)}{3} \]