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}+A_{2}+\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?