Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2015 based on Number Theory and Series.
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.
Series
Theory of Equations
Number Theory
But try the problem first...
Answer: is 648.
AIME, 2015, Question 1
Elementary Number Theory by David Burton
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.
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