If $|x| < 1$,then in the expansion of $(1 + 2x + 3x^2 + 4x^3 + ....)^{1/2}$,the coefficient of $x^n$ is

$1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots$ is equal to

The expansion of $(1+x+x^2)^{-3/2}$ in powers of $x$ is valid if

The $13^{\text{th}}$ term in the expansion of $(1-4x)^{-4}$ is

The binomial expansion $(7+3x)^{-2/5}$ is valid for all $x$ in the interval $\left(\frac{-7}{3}, \frac{7}{3}\right)$. If the $4^{th}$ term of its expansion is $kx^3$,then the value of $(7^{12/5}k)$ is:

The formula $(a + b)^m = a^m + ma^{m-1}b + \frac{m(m - 1)}{1 \cdot 2}a^{m - 2}b^2 + \dots$ holds when