If n is an even positive integer, then the co | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For the expansion $(1+x)^n$, the greatest coefficient occurs at the middle term$(s)$. Since $n$ is even, the greatest coefficient is at the term $

Vedclass · Accepted Answer

$\frac{n}{n+2} < x < \frac{n+2}{n}$

Vedclass · Answer

(A) For the expansion $(1+x)^n$, the greatest coefficient occurs at the middle term$(s)$. Since $n$ is even, the greatest coefficient is at the term $T_{n/2+1}$.For the greatest term $T_{r+1}$ in the expansion of $(1+x)^n$, we have the condition $\frac{n-r+1}{r} x \ge 1$ and $\frac{n-r+1}{r+1} x \le 1$.For the greatest term to be the one with the greatest coefficient, we set $r = n/2$.Substituting $r = n/2$ into the inequality $\frac{n-r+1}{r} x > 1$ and $\frac{n-r+1}{r+1} x $\frac{n - n/2 + 1}{n/2} x > 1$ $\Rightarrow \frac{n/2 + 1}{n/2} x > 1$ $\Rightarrow \frac{n+2}{n} x > 1$ $\Rightarrow x > \frac{n}{n+2}$.$\frac{n - n/2 + 1}{n/2 + 1} x Thus, the condition is $\frac{n}{n+2} < x < \frac{n+2}{n}$.

If $n$ is an even positive integer, then the condition that the greatest term in the expansion of $(a+x)^{n}$ may also have the greatest coefficient is

Similar Questions

Let $\alpha > 0$ be the smallest number such that the expansion of $(x^{2/3} + 2x^{-3})^{30}$ has a term $\beta x^{-\alpha}$,where $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $.............$.

$A$ possible value of $x$,for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$,is:

The coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1)^{15}$ is equal to:

The middle term in the expansion of $\left(4x^3 - \frac{15}{4x}\right)^8$ is

Arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then,the number of terms in the expansion having integer power of $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module