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


But try the problem first…

Answer: is 722.

Source
Suggested Reading

AIME I, 2015, Question 1

Elementary Number Theory by Sierpinsky

Try with Hints


First hint

A = \((1\times2)+(3\times4)\)

\(+(5\times6)+…+(35\times36)+(37\times38)+39\)

Second Hint

B=\(1+(2\times3)+(4\times5)\)

\(+(6\times7)+…+(36\times37)+(38\times39)\)

Final Step

B-A=\(-38+(2\times2)+(2\times4)\)

\(+(2\times6)+…+(2\times36)+(2\times38)\)

=722.

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