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# Theory of Equations | AIME I, 2015 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Theory of Equations.

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

But try the problem first…

Source

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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