Question 3
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives $$
\begin{aligned}&\binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1\\&+\binom{1000}{2}(0.2)^2+\cdots\\&+\binom{1000}{1000}(0.2)^{1000}\\&=A_0+A_1+\cdots+A_{1000},\end{aligned}
$$ where $A_k=\binom{1000}{k}(0.2)^k$ for $k=0,$ $1,$ $2,$ $\ldots,$ $1000.$ For which $k$ is $A_k$ the largest?
\begin{aligned}&\binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1\\&+\binom{1000}{2}(0.2)^2+\cdots\\&+\binom{1000}{1000}(0.2)^{1000}\\&=A_0+A_1+\cdots+A_{1000},\end{aligned}
$$ where $A_k=\binom{1000}{k}(0.2)^k$ for $k=0,$ $1,$ $2,$ $\ldots,$ $1000.$ For which $k$ is $A_k$ the largest?