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**

**S**eries

Equations

Number Theory

## Check the Answer

But try the problem first…

Answer: is 722.

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.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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