11. Binomial Theorem & Mathematical Induction question bank (Maths)

\[
\begin{aligned}
&\text{The coefficient of } x^{49} \text{ in the expansion of } (x-1)\left(x-\frac{1}{2}\right)\left(x-\frac{1}{2^2}\right)\ldots\left(x-\frac{1}{2^{49}}\right) \text{ is equal to:}\\
&

\[
\begin{array}{l}
\text{The largest real value for } x \text{ such that } 
\sum_{k=0}^{4} 
\Bigg( 
\frac{5^{4-k}}{(4-k)!} \cdot \frac{x^{k}}{k!} 
\Bigg)
= \frac{8}{3} \text{ is:} \\

\end{array}
\]