The coefficient of $x^3$ in the expansion of $\frac{(1 + 3x)^2}{1 - 2x}$ is:

The sum to infinite terms of the series $\frac{3}{10}+\frac{3 \cdot 7}{10 \cdot 15}+\frac{3 \cdot 7 \cdot 11}{10 \cdot 15 \cdot 20}+\ldots$ to $\infty$ is

The coefficient of ${x^r}$ in the expansion of ${(1 - 2x)^{-1/2}}$ is

If $(2-5x)^{-1/5} = a_0 + a_1x + a_2x^2 + \ldots$,then $\frac{a_1}{a_2} = $

The sum of the infinite series $1+\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{3 \cdot 6 \cdot 9 \cdot 12}+\ldots$ is equal to

If $x = \frac{2 \cdot 5}{2! \cdot 3} + \frac{2 \cdot 5 \cdot 7}{3! \cdot 3^2} + \frac{2 \cdot 5 \cdot 7 \cdot 9}{4! \cdot 3^3} + \ldots$,then $x^2 + 8x + 8 = $