Find the $r^{\text {th }}$ term from the end in the expansion of $(x+a)^{n}$
There are $(n+1)$ terms in the expansion of $(x+a)^{n}$. Observing the terms we can say that the first term from the end is the last term, i.e., $(n+1)^{\text {th }}$ term of the expansion and $n+1=(n+1)-(1-1) .$
The second term from the end is the $n^{\text {th }}$ term of the expansion, and $n=(n+1)-(2-1) .$
The third term from the end is the $(n-1)^{\text {th }}$ term of the expansion and $n-1=(n+1)-(3-1)$ and so on.
Thus $r^{th}$ term from the end will be term number $(n+1)-(r-1)=(n-r+2)$ of the expansion. And the $(n-r+2)^{ th }$ term is $^{n} C _{n-r+1} x^{r-1} a^{n-r+1}$
The second, third and fourth terms in the binomial expansion $(x+a)^n$ are $240,720$ and $1080,$ respectively. Find $x, a$ and $n$
The coefficient of $x^9$ in the expansion of $(1+x)\left(1+x^2\right)\left(1+x^3\right) \ldots . .\left(1+x^{100}\right)$ is
If the $6^{th}$ term in the expansion of the binomial ${\left[ {\sqrt {{2^{\log (10 - {3^x})}}} + \sqrt[5]{{{2^{(x - 2)\log 3}}}}} \right]^m}$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first, third and fifth terms of an $A.P$. (the symbol log stands for logarithm to the base $10$), then $x = $
In the expansion of ${(1 + x + {x^3} + {x^4})^{10}},$ the coefficient of ${x^4}$ is
Arrange the expansion of $\left(x^{1 / 2}+\frac{1}{2 x^{1 / 4}}\right)^n$ in decreasing powers of $x$.Suppose the coeff icients of the first three terms form an arithmetic progression. Then, the number of terms in the expansion having integer power of $x$ is