Difference between revisions of "2015 AIME I Problems/Problem 1"

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(Problem)
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==Problem==
 
==Problem==
  
The expressions <math>A</math> = <math> 1 \times 2 + 3 \times 4 + 5 \times 6 + *** + 37 \times 38 + 39 </math> and <math>B</math> = <math> 1 + 2 \times 3 + 4 \times 5 + *** + 36 \times 37 + 38 \times 39 </math> are obtained by writing multiplication and addition operators in an alternating pattern between successive integers.  Find the positive difference between integers <math>A</math> and <math>B</math>.
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The expressions <math>A</math> = <math> 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39 </math> and <math>B</math> = <math> 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39 </math> are obtained by writing multiplication and addition operators in an alternating pattern between successive integers.  Find the positive difference between integers <math>A</math> and <math>B</math>.
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==Solution==
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We see that
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<math>A=(1\times 2)+(3\times 4)+(5\times 6)+\cdots +(35\times 36)+(37\times 38)+39</math>
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and
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<math>B=1+(2\times 3)+(4\times 5)+(6\times 7)+\cdots +(36\times 37)+(38\times 39)</math>.
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Therefore,
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<math>B-A=-38+(2\times 2)+(2\times 4)+(2\times 6)+\cdots +(2\times 36)+(2\times 38)</math>
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<math>=-38+4\times (1+2+3+\cdots+19)</math>
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<math>=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.</math>
  
 
== See also ==
 
== See also ==

Revision as of 16:58, 20 March 2015

Problem

The expressions $A$ = $1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39$ and $B$ = $1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.

Solution

We see that

$A=(1\times 2)+(3\times 4)+(5\times 6)+\cdots +(35\times 36)+(37\times 38)+39$

and

$B=1+(2\times 3)+(4\times 5)+(6\times 7)+\cdots +(36\times 37)+(38\times 39)$.

Therefore,

$B-A=-38+(2\times 2)+(2\times 4)+(2\times 6)+\cdots +(2\times 36)+(2\times 38)$

$=-38+4\times (1+2+3+\cdots+19)$

$=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.$

See also

2015 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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