Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Theory of Equations.
Theory of Equations – AIME I, 2015
The expressions A=\(1\times2+3\times4+5\times6+…+37\times38+39\)and B=\(1+2\times3+4\times5+…+36\times37+38\times39\) 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 722
- is 250
- is 840
- cannot be determined from the given information
Key Concepts
Series
Equations
Number Theory
Check the Answer
Answer: is 722.
AIME I, 2015, Question 1
Elementary Number Theory by Sierpinsky
Try with Hints
A = \((1\times2)+(3\times4)\)
\(+(5\times6)+…+(35\times36)+(37\times38)+39\)
B=\(1+(2\times3)+(4\times5)\)
\(+(6\times7)+…+(36\times37)+(38\times39)\)
B-A=\(-38+(2\times2)+(2\times4)\)
\(+(2\times6)+…+(2\times36)+(2\times38)\)
=722.
Other useful links
- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s