Difference between revisions of "2002 AIME I Problems/Problem 7"
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(I looked it up on Wikipedia, and corrected the binomial expansion.) |
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The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers <math>x,y</math> and <math>r</math> with <math>|x|>|y|</math>, | The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers <math>x,y</math> and <math>r</math> with <math>|x|>|y|</math>, | ||
| − | <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}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots</cmath> |
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>? | 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 == | ||
| − | {{ | + | <math>1^n</math> will always be 1, so we can cut out those terms, and we now have |
| + | |||
| + | <cmath>10^{2860}+\dfrac{10}{7}10^{858}+\dfrac{15}{49}10^{-1144}+\cdots</cmath> | ||
| + | |||
| + | Since the exponent in the 10 goes down extremely fast, we just need to consider the first few terms. Also, we can cut the <math>10^{2860}</math> out, so we need to find the first three digits after the decimal in | ||
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| + | <cmath>\dfrac{10}{7}10^{858}</cmath>. | ||
| + | |||
| + | Since the repeating decimal of <math>\dfrac{10}{7}</math> repeats every 6 digits, we can cut out a lot of 6's from 858 to get | ||
| + | |||
| + | <math>\dfrac{10}{7}</math>. | ||
| + | |||
| + | That is the same as <math>1+\dfrac{3}{7}</math>, and the first three digits after <math>\dfrac{3}{7}</math> are <math>\boxed{428}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2002|n=I|num-b=6|num-a=8}} | {{AIME box|year=2002|n=I|num-b=6|num-a=8}} | ||
Revision as of 09:18, 5 May 2008
Problem
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers 
and 
with 
,
![\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]](http://latex.artofproblemsolving.com/4/3/c/43c3768ec432dd7b10015400b663d91b7ef8e6aa.png)
What are the first three digits to the right of the decimal point in the decimal representation of 
?
Solution
will always be 1, so we can cut out those terms, and we now have
![\[10^{2860}+\dfrac{10}{7}10^{858}+\dfrac{15}{49}10^{-1144}+\cdots\]](http://latex.artofproblemsolving.com/6/2/a/62a60868f3c93dbae4d80fbd9c1cfe55dcb1d72e.png)
Since the exponent in the 10 goes down extremely fast, we just need to consider the first few terms. Also, we can cut the 
out, so we need to find the first three digits after the decimal in
![\[\dfrac{10}{7}10^{858}\]](http://latex.artofproblemsolving.com/d/5/3/d5307a89b95cd1016bba711891618a1b63d7a300.png)
.
Since the repeating decimal of 
repeats every 6 digits, we can cut out a lot of 6's from 858 to get
.
That is the same as 
, and the first three digits after 
are 
.
See also
| 2002 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
