Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2015 based on Number Theory and Series.
Number Theory and Series – AIME 2015
The expressions A = \((1 \times 2)+(3 \times 4)+….+(35 \times 36)+37\) and B = \(1+(2 \times 3)+(4 \times 5)+….+(36 \times 37)\) are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers A and B.
- is 107
- is 648
- is 840
- cannot be determined from the given information
Key Concepts
Series
Theory of Equations
Number Theory
Check the Answer
But try the problem first…
Answer: is 648.
AIME, 2015, Question 1
Elementary Number Theory by David Burton
Try with Hints
First hint
B-A=\(-36+(2 \times 3)+….+(2 \times 36)\)
Second Hint
=\(-36+4 \times (1+2+3+….+18)\)
Final Step
=\(-36+(4 \times \frac{18 \times 19}{2})\)=648.
Other useful links
- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s
Related Program
Math Olympiad Program… taught by Olympians and researchers
