Difference between revisions of "2002 AIME I Problems/Problem 7"

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(Problem)
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<cmath>(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y\cdots</cmath>
 
<cmath>(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y\cdots</cmath>
  
What are the first three digits to the right of the decimal point in the decimal representation of <math>(10^{2002}+1)^{\dfrac{10}{7}}</math>?
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What are the first three digits to the right of the decimal point in the decimal representation of <math>(10^{2002}+1)^{\frac{10}{7}}</math>?
  
 
== Solution ==
 
== Solution ==

Revision as of 15:34, 25 September 2007

Problem

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,

\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y\cdots\]

What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$?

Solution

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See also