700 Followers Problem!

Algebra Level 5

Expanding (1+0.2)1000(1+0.2)^{1000} by the Binomial theorem gives

(10000)(0.2)0+(10001)(0.2)1+(10002)(0.2)2++(10001000)(0.2)1000{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} =A0+A1+A2++A1000= A_0 + A_1 + A_2 + \cdots + A_{1000}

Where Ak=(1000k)(0.2)kA_k = {1000 \choose k}(0.2)^k for k=0,1,2,,1000k = 0,1,2,\ldots,1000.

For which kk_{}^{} is AkA_k^{} the largest?


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Image Credit: Wikimedia Empirical Rule by Dan Kernler
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