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>. | ||
| − | ==Solution== | + | ==Solution 1== |
We see that | We see that | ||
| Line 20: | Line 20: | ||
<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)== | ||
| + | |||
| + | 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: | ||
| + | |||
| + | |||
| + | <math>2, 12, 30, 56, 90...(39)</math> | ||
| + | |||
| + | <math>6, 18, 26, 34...</math> | ||
| + | |||
| + | <math>8, 8, 8...</math> | ||
| + | |||
| + | and | ||
| + | |||
| + | <math>(1) 6, 20, 42, 72...</math> | ||
| + | |||
| + | <math>14, 22, 30...</math> | ||
| + | |||
| + | <math>8, 8, 8...</math> | ||
| + | |||
| + | |||
| + | 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: | ||
| + | |||
| + | <math>(19choose1)/times 2</math> | ||
== See also == | == See also == | ||
Revision as of 13:53, 25 March 2020
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 .
Solution 1
We see that
and
.
Therefore,
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:
and
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:
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. 
