Find the $r^{\text{th}}$ term from the end in the expansion of $(x+a)^{n}$.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
The expansion of $(x+a)^{n}$ contains $(n+1)$ terms.
To find the $r^{\text{th}}$ term from the end,we observe the pattern:
$1^{\text{st}}$ term from the end is the $(n+1)^{\text{th}}$ term.
$2^{\text{nd}}$ term from the end is the $n^{\text{th}}$ term.
$3^{\text{rd}}$ term from the end is the $(n-1)^{\text{th}}$ term.
In general,the $r^{\text{th}}$ term from the end is the $(n+1)-(r-1) = (n-r+2)^{\text{th}}$ term from the beginning.
The general term of $(x+a)^{n}$ is given by $T_{k+1} = ^{n}C_{k} x^{n-k} a^{k}$.
For the $(n-r+2)^{\text{th}}$ term,we set $k+1 = n-r+2$,which gives $k = n-r+1$.
Substituting $k = n-r+1$ into the general term formula:
$T_{n-r+2} = ^{n}C_{n-r+1} x^{n-(n-r+1)} a^{n-r+1} = ^{n}C_{n-r+1} x^{r-1} a^{n-r+1}$.

Explore More

Similar Questions

The maximum value of the term independent of $t$ in the expansion of $\left( tx^{\frac{1}{5}} + \frac{(1-x)^{\frac{1}{10}}}{t} \right)^{10}$ where $x \in (0, 1)$ is

If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1 + ax + bx^2)(1 - 2x)^{18}$ in powers of $x$ are both zero,then $(a, b)$ is equal to

If the coefficient of the $3^{\text{rd}}$ term from the beginning in the expansion of $\left(ax^2 - \frac{8}{bx}\right)^9$ is equal to the coefficient of the $3^{\text{rd}}$ term from the end in the expansion of $\left(ax - \frac{2}{bx^2}\right)^9$,then the relation between $a$ and $b$ is:

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+ax)^9$ are the same, then the value of $a$ is

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo