Explanation

(d)
Let $S = 1 + 3x + 6x^{2} + 10x^{3} + ...\infty$ …(i)
$xS = x + 3x^{2} + 6x^{3} + ...\infty$ …(ii)
On subtracting Eq. (ii) from Eq. (i), we get
$S(1 - x) = 1 + 2x + 3x^{2} + 4x^{3} + ...\infty$ …(iii)
$\Rightarrow$ $x(1 - x)S = x + 2x^{2} + 3x^{3} + ...\infty$ …(iv)
Again, subtracting Eq. (iv) from Eq. (iii), we get
$S\left\lbrack (1 - x) - x(1 - x) \right\rbrack = (1 + x + x^{2} + x^{3} + ...\infty)$
$\Rightarrow S\left\lbrack (1 - x)(1 - x) \right\rbrack = \frac{1}{1 - x}$
$\Rightarrow S = \frac{1}{(1 - x)^{3}}$