Question

Sol. [Key Concept: $\sum(1 + 2 + \ldots + k)x^{k - 1}$: generating function gives $\frac{1}{(1 - x)^{3}}$ behaviour.]

$${S_{n} = 1 + (1 + 2)x + (1 + 2 + 3)x^{2} + \ldots + (1 + 2 + 3 + \ldots n)x^{n - 1} }{xS_{n} = x + (1 + 2)x^{2} + \ldots + (1 + 2 + \ldots n - 1)x^{n - 1} + (1 + 2 + \ldots n)x^{n} }{(1 - x)S_{n} = 1 + 2x + 3x^{2} + 4x^{3} + \ldots + nx^{n - 1} - (1 + 2 + 3 + \ldots + n)x^{n} }{x(1 - x)S_{n} = x + 2x^{2} + 3x^{3} + \ldots + (n - 1)x^{n - 1} + nx^{n} - (1 + 2 + 3 + \ldots n)x^{n + 1} }{(1 - x)^{2}S_{n} = 1 + x + x^{2} + \ldots x^{n - 1} - nx^{n} - (1 + 2 + 3 + \ldots n)x^{n}(1 - x)}$$

$$= \frac{1 - x^{n}}{1 - x} - nx^{n} - \frac{n(n + 1)}{2}x^{n}(1 - x)$$

$$\Rightarrow \ S_{n} = \frac{1 - x^{n}}{(1 - x)^{3}} - \frac{{nx}^{n}}{(1 - x)^{2}} - \frac{n(n + 1)x^{n}}{2(1 - x)}$$