2015 AIME I Problems/Problem 1: Difference between revisions

Revision as of 05:04, 2 January 2022

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 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:

$2, 12, 30, 56, 90...(39)$

$6, 18, 26, 34...$

$8, 8, 8...$

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 2: / Line 2: @@
 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>.
-==Solution==
+==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==

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