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Difference between revisions of "1962 AHSME Problems/Problem 20"

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==Problem=
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==Problem==
  
 
The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:  
 
The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:  
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==Solution==
 
==Solution==
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"Unsolved"
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If the angles are in an arithmetic progression, they can be expressed as
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<math>a</math>, <math>a+n</math>, <math>a+2n</math>, <math>a+3n</math>, and <math>a+4n</math> for some real numbers <math>a</math> and <math>n</math>.
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Now we know that the sum of the degree measures of the angles of a pentagon is <math>180(5-2)=540</math>.
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Adding our expressions for the five angles together, we get <math>5a+10n=540</math>.
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We now divide by 5 to get <math>a+2n=108</math>. It so happens that <math>a+2n</math> is one of the angles we defined earlier, so that angle must have a measure of <math>\boxed{108\textbf{ (A)}}</math>.
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(In fact, for any arithmetic progression with an odd number of terms,
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the middle term is equal to the average of all the terms.)
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==Solution 2==
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If we write the five terms as <math>a</math>, <math>a - n</math>, <math>a - 2n</math>, <math>a + n</math> and <math>a + 2n</math>, we can see that adding them up, we get <math>5a = 540</math> through this, we can see that <math>a = 108</math>, <math>\fbox{\textbf{(A)}}</math>
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==See Also==
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{{AHSME 40p box|year=1962|before=Problem 19|num-a=21}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 19:00, 29 October 2020

Problem

The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:

$\textbf{(A)}\ 108\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 54\qquad\textbf{(E)}\ 36$

Solution

If the angles are in an arithmetic progression, they can be expressed as $a$, $a+n$, $a+2n$, $a+3n$, and $a+4n$ for some real numbers $a$ and $n$. Now we know that the sum of the degree measures of the angles of a pentagon is $180(5-2)=540$. Adding our expressions for the five angles together, we get $5a+10n=540$. We now divide by 5 to get $a+2n=108$. It so happens that $a+2n$ is one of the angles we defined earlier, so that angle must have a measure of $\boxed{108\textbf{ (A)}}$. (In fact, for any arithmetic progression with an odd number of terms, the middle term is equal to the average of all the terms.)

Solution 2

If we write the five terms as $a$, $a - n$, $a - 2n$, $a + n$ and $a + 2n$, we can see that adding them up, we get $5a = 540$ through this, we can see that $a = 108$, $\fbox{\textbf{(A)}}$

See Also

1962 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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