2015 AIME I Problems/Problem 1: Difference between revisions

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

@@ Line 1: / Line 1: @@
 ==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>.
+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==
+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
+<math>A=(1\times 2)+(3\times 4)+(5\times 6)+\cdots +(35\times 36)+(37\times 38)+39</math>
+and
+<math>B=1+(2\times 3)+(4\times 5)+(6\times 7)+\cdots +(36\times 37)+(38\times 39)</math>.
+Therefore,
+<math>B-A=-38+(2\times 2)+(2\times 4)+(2\times 6)+\cdots +(2\times 36)+(2\times 38)</math>
+<math>=-38+4\times (1+2+3+\cdots+19)</math>
+<math>=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.</math>
+==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:
+<cmath>2, 12, 30, 56, 90...(39)</cmath>
+<cmath>10, 18, 26, 34...</cmath>
+<cmath>8, 8, 8...</cmath>
+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 ==
 {{AIME box|year=2015|n=I|before=First Problem|num-a=2}}
 {{MAA Notice}}
-[[Category:Introductory Geometry Problems]]
+[[Category:Introductory Algebra Problems]]

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