AIME 1991 #3 — Binomial Expansion (1.2)^1000 Largest Term
2200ExpertBinomial TheoremCombinatorics
AIME · 1991 · Problem 3
Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives
<center>${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$</center>
<center>$= A_0 + A_1 + A_2 + \cdots + A_{1000},$</center>
where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k_{}^{}$ is $A_k^{}$ the largest?