Difference between revisions of "2015 AIME I Problems/Problem 1"
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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>. | 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>. | ||
| + | |||
| + | ==Video Solution For Problems 1-3== | ||
| + | https://www.youtube.com/watch?v=5HAk-6qlOH0 | ||
==Solution 1== | ==Solution 1== | ||
| + | We have <cmath>|A-B|=|1+3(4-2)+5(6-4)+ \cdots + 37(38-36)-39(1-38)|</cmath><cmath>\implies |2(1+3+5+7+ \cdots +37)-1-39(37)|</cmath><cmath>\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\implies \boxed{722}</cmath> | ||
| + | |||
| + | ==Solution 2== | ||
We see that | We see that | ||
| Line 21: | Line 27: | ||
<math>=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.</math> | <math>=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.</math> | ||
| − | ==Solution | + | ==Solution 3 (slower solution)== |
For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations. | For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations. | ||
| Line 28: | Line 34: | ||
| − | < | + | <cmath>2, 12, 30, 56, 90...(39)</cmath> |
| − | < | + | <cmath>6, 18, 26, 34...</cmath> |
| − | < | + | <cmath>8, 8, 8...</cmath> |
and | and | ||
| − | < | + | <cmath>(1) 6, 20, 42, 72...</cmath> |
| + | <cmath>14, 22, 30...</cmath> | ||
| − | < | + | <cmath>8, 8, 8...</cmath> |
| − | |||
| + | Then we use Newton's Little Formula for the sum of <math>n</math> terms in a sequence. | ||
| − | + | Notice that there are <math>19</math> terms in each sequence, plus the tails of <math>39</math> and <math>1</math> on the first and second equations, respectively. | |
| − | + | So, | |
| − | + | <cmath>2\binom{19}{1}+10\binom{19}{2}+8\binom{19}{3}+1</cmath> | |
| − | + | <cmath>6\binom{19}{1}+14\binom{19}{2}+8\binom{19}{3}+39</cmath> | |
| − | + | Subtracting <math>A</math> from <math>B</math> gives: | |
| − | + | <cmath>4\binom{19}{1}+4\binom{19}{2}-38</cmath> | |
| − | + | Which unsurprisingly gives us <math>\boxed{722}.</math> | |
| − | + | -jackshi2006 | |
== See also == | == See also == | ||
Latest revision as of 23:26, 8 January 2023
Contents
Problem
The expressions 
= 
and 
= 
are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers and .
Video Solution For Problems 1-3
https://www.youtube.com/watch?v=5HAk-6qlOH0
Solution 1
We have ![\[|A-B|=|1+3(4-2)+5(6-4)+ \cdots + 37(38-36)-39(1-38)|\]](http://latex.artofproblemsolving.com/a/a/0/aa00ddd78b4ebb062abbcea06b77955873953461.png)
![\[\implies |2(1+3+5+7+ \cdots +37)-1-39(37)|\]](http://latex.artofproblemsolving.com/9/2/c/92cefabadfba0fc9331e69a9b8f162e2bdd8a867.png)
![\[\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\implies \boxed{722}\]](http://latex.artofproblemsolving.com/7/9/a/79a2b444af5b633151febc53b992e41203462fae.png)
Solution 2
We see that
and
.
Therefore,
Solution 3 (slower solution)
For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations.
We write down the pairs of numbers after multiplication and solve each layer:
![\[2, 12, 30, 56, 90...(39)\]](http://latex.artofproblemsolving.com/c/9/3/c93c012e6a889359a0368b96b7fe18a8ece0671f.png)
![\[6, 18, 26, 34...\]](http://latex.artofproblemsolving.com/8/6/8/868a5a54947c58499046ca955215ac26fef9062e.png)
![\[8, 8, 8...\]](http://latex.artofproblemsolving.com/c/1/3/c1350211e713e87ff5eec2b726ee6ec799943179.png)
and
![\[(1) 6, 20, 42, 72...\]](http://latex.artofproblemsolving.com/4/9/b/49bcb557f69b2c058d46a8074e2c74c7161627ac.png)
![\[14, 22, 30...\]](http://latex.artofproblemsolving.com/c/c/f/ccfcc44e95127c3d5d01936838c9bcd081cb656b.png)
Then we use Newton's Little Formula for the sum of 
terms in a sequence.
Notice that there are 
terms in each sequence, plus the tails of 
and 
on the first and second equations, respectively.
So,
![\[2\binom{19}{1}+10\binom{19}{2}+8\binom{19}{3}+1\]](http://latex.artofproblemsolving.com/1/0/4/104d1af766a33e256386fe5264c2a71c936cc06b.png)
![\[6\binom{19}{1}+14\binom{19}{2}+8\binom{19}{3}+39\]](http://latex.artofproblemsolving.com/4/b/f/4bf46b537cec3663375717de9aa4b11b824d0e33.png)
Subtracting from gives:
![\[4\binom{19}{1}+4\binom{19}{2}-38\]](http://latex.artofproblemsolving.com/b/6/7/b67e62919b2b8b5ae8ec68edaeac8d40beddab40.png)
Which unsurprisingly gives us 
-jackshi2006
See also
| 2015 AIME I (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
