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AIME I Algebra Arithmetic Calculus Math Olympiad USA Math Olympiad

Number and Series | Number Theory | AIME I, 2015

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Probability Biased and Unbiased.

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


Answer: is 648.

AIME, 2015, Question 1

Elementary Number Theory by David Burton

Try with Hints


B-A=\(-36+(2 \times 3)+….+(2 \times 36)\)

=\(-36+4 \times (1+2+3+….+18)\)

=\(-36+(4 \times \frac{18 \times 19}{2})\)=648.

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