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

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(→‎Solution 2 (slower solution))
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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 19 terms in each sequence, plus the tails of 39 and 1 on the first and second equations, respectively.
  
−
So:
+
So: (this is as far as my latex knowledge goes)
  
−
<math>(19choose1)/times 2</math>
+
((19 choose 1) x 2) + ((19 choose 2) x 10) + ((19 choose 3) x 8) + 1
 +
 
 +
((19 choose 1) x 6) + ((19 choose 2) x 14) + ((19 choose 3) x 8) + 39
 +
 
 +
Subtracting A from B gives:
 +
 
 +
((19 choose 1) x 4) + ((19 choose 2) x 4) - 38
 +
 
 +
Which unsurprisingly gives us <math>/boxed{722}.</math>
  
 
== See also ==
 
== See also ==

Revision as of 14:08, 25 March 2020

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 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 2 (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: (this is as far as my latex knowledge goes)

((19 choose 1) x 2) + ((19 choose 2) x 10) + ((19 choose 3) x 8) + 1

((19 choose 1) x 6) + ((19 choose 2) x 14) + ((19 choose 3) x 8) + 39

Subtracting A from B gives:

((19 choose 1) x 4) + ((19 choose 2) x 4) - 38

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

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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