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Difference between revisions of "1971 AHSME Problems/Problem 24"

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== Solution ==
 
== Solution ==
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<math>\boxed{\textbf{(D) }\frac{n^2-3n+2}{4n-2}}</math>.
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The first row has <math>1</math> one, and all the other rows have <math>2</math> ones. Thus, the total number of ones in the first <math>n</math> rows is <math>1+2(n-1)=2n-1</math>. The total number of elements in the first <math>n</math> rows is <math>1+2+3+\ldots+n</math>, which is the <math>n</math>th [[triangular number]], <math>\tfrac{n(n+1)}2</math>. Thus, the number of elements excluding ones is <math>\tfrac{n(n+1)}2-(2n-1)</math>. Thus, our desired quotient is <math>\frac{\tfrac{n(n+1)}2-(2n-1)}{2n-1} = \frac{n^2+n-2(2n-1)}{2(2n-1)} = \boxed{\textbf{(D) }\frac{n^2-3n+2}{4n-2}}</math>.
  
 
== See Also ==
 
== See Also ==
 
{{AHSME 35p box|year=1971|num-b=23|num-a=25}}
 
{{AHSME 35p box|year=1971|num-b=23|num-a=25}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:47, 6 August 2024

Problem

[asy] label("$1$",(0,0),S); label("$1$",(-1,-1),S); label("$1$",(-2,-2),S); label("$1$",(-3,-3),S); label("$1$",(-4,-4),S); label("$1$",(1,-1),S); label("$1$",(2,-2),S); label("$1$",(3,-3),S); label("$1$",(4,-4),S); label("$2$",(0,-2),S); label("$3$",(-1,-3),S); label("$3$",(1,-3),S); label("$4$",(-2,-4),S); label("$4$",(2,-4),S); label("$6$",(0,-4),S); label("etc.",(0,-5),S); //Credit to chezbgone2 for the diagram[/asy]

Solution

The first row has $1$ one, and all the other rows have $2$ ones. Thus, the total number of ones in the first $n$ rows is $1+2(n-1)=2n-1$. The total number of elements in the first $n$ rows is $1+2+3+\ldots+n$, which is the $n$th triangular number, $\tfrac{n(n+1)}2$. Thus, the number of elements excluding ones is $\tfrac{n(n+1)}2-(2n-1)$. Thus, our desired quotient is $\frac{\tfrac{n(n+1)}2-(2n-1)}{2n-1} = \frac{n^2+n-2(2n-1)}{2(2n-1)} = \boxed{\textbf{(D) }\frac{n^2-3n+2}{4n-2}}$.

See Also

1971 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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