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2015 AIME I Problems/Problem 1: Difference between revisions

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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
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<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 2 (slower 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:




<math>2, 12, 30, 56, 90...(39)</math>
<cmath>2, 12, 30, 56, 90...(39)</cmath>


<math>6, 18, 26, 34...</math>
<cmath>10, 18, 26, 34...</cmath>


<math>8, 8, 8...</math>
<cmath>8, 8, 8...</cmath>


and
and


<math>(1) 6, 20, 42, 72...</math>
<cmath>(1) 6, 20, 42, 72...</cmath>
 
<cmath>14, 22, 30...</cmath>
<math>14, 22, 30...</math>
 
<math>8, 8, 8...</math>


<cmath>8, 8, 8...</cmath>


Then we use Newton's Little Formula for the sum of n terms in a sequence.


Then we use Newton's Little Formula for the sum of <math>n</math> terms in a sequence.


Notice that there are 19 terms in each sequence, plus the tails of 39 and 1 on the first and second equations, respectively.
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: (this is as far as my latex knowledge goes)
So,




((19 choose 1) x 2) + ((19 choose 2) x 10) + ((19 choose 3) x 8) + 1
<cmath>2\binom{19}{1}+10\binom{19}{2}+8\binom{19}{3}+1</cmath>




((19 choose 1) x 6) + ((19 choose 2) x 14) + ((19 choose 3) x 8) + 39
<cmath>6\binom{19}{1}+14\binom{19}{2}+8\binom{19}{3}+39</cmath>




Subtracting A from B gives:
Subtracting <math>A</math> from <math>B</math> gives:




((19 choose 1) x 4) + ((19 choose 2) x 4) - 38
<cmath>4\binom{19}{1}+4\binom{19}{2}-38</cmath>




Latest revision as of 20:19, 1 February 2025

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 $B$.

Video Solution For Problems 1-3

https://www.youtube.com/watch?v=5HAk-6qlOH0

Solution 1

We have \[|A-B|=|1+3(4-2)+5(6-4)+ \cdots + 37(38-36)-39(1-38)|\]\[\implies |2(1+3+5+7+ \cdots +37)-1-39(37)|\]\[\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\implies \boxed{722}\]

Solution 2

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,

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

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

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

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:


\[2, 12, 30, 56, 90...(39)\]

\[10, 18, 26, 34...\]

\[8, 8, 8...\]

and

\[(1) 6, 20, 42, 72...\] \[14, 22, 30...\]

\[8, 8, 8...\]


Then we use Newton's Little Formula for the sum of $n$ terms in a sequence.

Notice that there are $19$ terms in each sequence, plus the tails of $39$ and $1$ on the first and second equations, respectively.


So,


\[2\binom{19}{1}+10\binom{19}{2}+8\binom{19}{3}+1\]


\[6\binom{19}{1}+14\binom{19}{2}+8\binom{19}{3}+39\]


Subtracting $A$ from $B$ gives:


\[4\binom{19}{1}+4\binom{19}{2}-38\]


Which unsurprisingly gives us $\boxed{722}.$


-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

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