Binomial Theorem
Middle Term and Odd-Power Sum
nta_pyq_2023_jan
Grade None
Question:
Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $\left(2+\dfrac{1}{\sqrt{2}}\right)^{200}$. If $\dfrac{{}^{200}C_{99}\,K}{a}=\dfrac{2^\ell\,m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(\ell,n)$ is equal to:
(50, 51)
(51, 99)
(50, 101)
(51, 101)
Step-by-Step Solution
Key Concept: $K=2^{98}$. Middle term $a={}^{200}C_{100}\cdot2^{100}\cdot(1/\sqrt{2})^{100}={}^{200}C_{100}\cdot2^{50}$.
$(\ell,n)=(50,101)$.
Correct Answer: 3