Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 26"
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== Problem == | == Problem == | ||
| + | Let <math>n=1667</math>. Then the first nonzero digit in the decimal expansion of <math>\sqrt{n^2 + 1} - n</math> is | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ }4 \qquad \mathrm{(E) \ }5 </math></center> |
== Solution == | == Solution == | ||
| + | We only need to worry about the decimal expansion of <math>\sqrt{n^{2}+1}</math> since the <math>-1</math> will do nothing to the decimal digits. We have <math>(1667^{2}+1)^{1/2}</math>. Using the extended binomial theorem, we have that <math> \displaystyle {1/2\choose 0} (1667) + {1/2\choose 1} \left(\frac 1{1667}\right) + \cdots </math>. Now we only have to look at the second term since all the following terms will be too small to affect the first nonzero digit of the decimal expansion. We see that <math>\frac{1}{2 \cdot 1667}=.00029\ldots</math>. The answer is <math>2</math>. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] | ||
| + | |||
| + | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 19:31, 22 July 2006
Problem
Let 
. Then the first nonzero digit in the decimal expansion of 
is
Solution
We only need to worry about the decimal expansion of 
since the 
will do nothing to the decimal digits. We have 
. Using the extended binomial theorem, we have that 
. Now we only have to look at the second term since all the following terms will be too small to affect the first nonzero digit of the decimal expansion. We see that 
. The answer is 
.
