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Difference between revisions of "2002 AMC 12P Problems/Problem 11"

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== Problem ==
 
== Problem ==
−
How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
+
Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find
  
−
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math>
+
<cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}</cmath>
  
−
== Solution ==
+
<math>
−
If <math>\log_{b} 729 = n</math>, then <math>b^n = 729</math>. Since <math>729 = 3^6</math>, <math>b</math> must be <math>3</math> to some [[factor]] of 6. Thus, there are four (3, 9, 27, 729) possible values of <math>b \Longrightarrow \boxed{\mathrm{E}}</math>.
+
\text{(A) }\frac {4003}{2003}
 +
\qquad
 +
\text{(B) }\frac {2001}{1001}
 +
\qquad
 +
\text{(C) }\frac {4004}{2003}
 +
\qquad
 +
\text{(D) }\frac {4001}{2001}
 +
\qquad
 +
\text{(E) }2
 +
</math>
 +
 
 +
== Solution 1 ==
 +
We may write <math>\frac{1}{t_n}</math> as <math>\frac{2}{n(n+1)}</math> and do a partial fraction decomposition.
 +
Assume <math>\frac{2}{n(n+1)} = \frac{A_1}{n} + \frac{A_2}{n+1}</math>.
 +
 
 +
Multiplying both sides by <math>n(n+1)</math> gives <math>2 = A_1(n+1) + A_2(n) = (A_1 + A_2)n + A_1</math>.
 +
 
 +
Equating coefficients gives <math>A_1 = 2</math> and <math>A_1 + A_2 = 0</math>, so <math>A_2 = -2</math>. Therefore, <math>\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}</math>.
 +
 
 +
Now <math>\frac{1}{t_1} + \frac{1}{t_2} + ... + \frac{1}{t_{2002}} = 2((\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + ... + (\frac{1}{2002} - \frac{1}{2003})) = 2(1 - \frac{1}{2003}) = \boxed{\frac {4004}{2003} \text{(C) }}</math>.
 +
 
 +
Note: For the sake of completeness, I put the full derivation of the partial fraction decomposition of <math>\frac{2}{n(n+1)}</math> here. However, on the contest, the decomposition step would be much faster since it is so well-known.
  
 
== See also ==
 
== See also ==
−
{{AMC12 box|year=2000|num-b=6|num-a=8}}
+
{{AMC12 box|year=2002|ab=P|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:18, 10 March 2024

Problem

Let $t_n = \frac{n(n+1)}{2}$ be the $n$th triangular number. Find

\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}\]

$\text{(A) }\frac {4003}{2003} \qquad \text{(B) }\frac {2001}{1001} \qquad \text{(C) }\frac {4004}{2003} \qquad \text{(D) }\frac {4001}{2001} \qquad \text{(E) }2$

Solution 1

We may write $\frac{1}{t_n}$ as $\frac{2}{n(n+1)}$ and do a partial fraction decomposition. Assume $\frac{2}{n(n+1)} = \frac{A_1}{n} + \frac{A_2}{n+1}$.

Multiplying both sides by $n(n+1)$ gives $2 = A_1(n+1) + A_2(n) = (A_1 + A_2)n + A_1$.

Equating coefficients gives $A_1 = 2$ and $A_1 + A_2 = 0$, so $A_2 = -2$. Therefore, $\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}$.

Now $\frac{1}{t_1} + \frac{1}{t_2} + ... + \frac{1}{t_{2002}} = 2((\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + ... + (\frac{1}{2002} - \frac{1}{2003})) = 2(1 - \frac{1}{2003}) = \boxed{\frac {4004}{2003} \text{(C) }}$.

Note: For the sake of completeness, I put the full derivation of the partial fraction decomposition of $\frac{2}{n(n+1)}$ here. However, on the contest, the decomposition step would be much faster since it is so well-known.

See also

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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