If the coefficients of 2 nd , 3 rd and the 4 th terms in the expansion of $(1+x)^n$ are in AP, then the value of $n$ is
If $A$ and $B$ are coefficient of $x^n$ in the expansions of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, then $\frac{A}{B}$ equals to
If the middle term of $\left(\frac{1}{x}+x \sin x\right)^{10}$ is equal to $7 \frac{7}{8}$, then the value of $x$ is
The largest coefficient in the expansion of $(1+x)^{30}$ is .............. .
Largest coefficient in the expansion of $(1+x)^{30}={ }^{30} \mathrm{C}_{30 / 2}={ }^{30} \mathrm{C}_{15}$
The number of terms in the expansion of $(x+y+z)^n$ ............ .
Given expansion is $(x+y+z)^n=[x+(y+z)]^n$.
$$\begin{aligned} {[x+(y+z)]^n={ }^n C_0 x^n+{ }^n C_1 x^{n-1}(} & (y) z) \\ & +{ }^n C_2 x^{n-2}(y+z)^2+\ldots+{ }^n C_n(y+z)^n \end{aligned}$$
$$\begin{aligned} \therefore \text { Number of terms } & =1+2+3+\ldots+n+(n+1) \\ & =\frac{(n+1)(n+2)}{2} \end{aligned}$$