R^{text {th }} term from end in expansion of (x+a)^{n}

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7.Binomial Theorem

medium

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

Option A

Option B

Option C

Option D

Solution

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 $^{n} C _{n-r+1} x^{r-1} a^{n-r+1}$

Standard 11

Mathematics

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${\left( {{x^2} + \frac{2}{x}} \right)^{15}}$ is

hard

(JEE MAIN-2013)

The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$. Then $n=$

hard

(IIT-2013)

Let $K$ be the coefficient of $x^4$ in the expansion of $( 1 + x + ax^2) ^{10}$ . What is the value of $'a'$ that minimizes $K$ ?

normal

Let ${\left( {x + 10} \right)^{50}} + {\left( {x – 10} \right)^{50}} = {a_0} + {a_1}x + {a_2}{x^2} + …. + {a_{50}}{x^{50}}$ , for $x \in R$; then $\frac{{{a_2}}}{{{a_0}}}$ is equal to

hard

(JEE MAIN-2019)

Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0, n \in N$, be $376$. Then the coefficient of $x^4$ is $……$

hard

(JEE MAIN-2023)

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

Solution

Similar Questions

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${\left( {{x^2} + \frac{2}{x}} \right)^{15}}$ is

The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$. Then $n=$

Let $K$ be the coefficient of $x^4$ in the expansion of $( 1 + x + ax^2) ^{10}$ . What is the value of $'a'$ that minimizes $K$ ?

Let ${\left( {x + 10} \right)^{50}} + {\left( {x – 10} \right)^{50}} = {a_0} + {a_1}x + {a_2}{x^2} + …. + {a_{50}}{x^{50}}$ , for $x \in R$; then $\frac{{{a_2}}}{{{a_0}}}$ is equal to

Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0, n \in N$, be $376$. Then the coefficient of $x^4$ is $……$

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