2015 AIME I Problems/Problem 1
Contents
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 .
Video Solution For Problems 1-3
https://www.youtube.com/watch?v=5HAk-6qlOH0
Solution 1
We have
Solution 2
We see that
and
.
Therefore,
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:
and
Then we use Newton's Little Formula for the sum of terms in a sequence.
Notice that there are terms in each sequence, plus the tails of and on the first and second equations, respectively.
So,
Subtracting from gives:
Which unsurprisingly gives us
-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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