# Number and Series | Number Theory | AIME I, 2015

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

AIME, 2015, Question 1

Elementary Number Theory by David Burton

## Try with Hints

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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