Difference between revisions of "1991 AIME Problems/Problem 3"

Revision as of 18:59, 20 April 2007

Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives

${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$

$= A_0 + A_1 + A_2 + \cdots + A_{1000},$

where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$

For which $k_{}^{}$ is $A_k^{}$ the largest?

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 == Problem ==
 Expanding <math>(1+0.2)^{1000}_{}</math> by the binomial theorem and doing no further manipulation gives
-<center><math>{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}</math></center>
-<center><math>= A_0 + A_1 + A_2 + \cdots + A_{1000},</math></center>
+<math>{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}</math></center>
-where <math>A_k = {1000 \choose k}(0.2)^k</math> for <math>k = 0,1,2,\ldots,1000</math>. For which <math>k_{}^{}</math> is <math>A_k^{}</math> the largest?
+<center><math>= A_0 + A_1 + A_2 + \cdots + A_{1000},</math>
+where <math>A_k = {1000 \choose k}(0.2)^k</math> for <math>k = 0,1,2,\ldots,1000</math>
+For which <math>k_{}^{}</math> is <math>A_k^{}</math> the largest?
 == Solution ==

1991 AIME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AIME Problems and Solutions