jmc

Question

The expressions A=1 2 + 3 4 + 5 6 + + 37 38 + 39and B = 1 + 2 3 + 4 5 + + 36 37 + 38 39are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers A and B.

Accepted Answer

Instead of computing A and B separately, we can write a simple expression for A-B, as follows: aligned A - B &= (1 2 + 3 4 + 5 6 + + 37 38 + 39) - (1 + 2 3 + 4 5 + + 36 37 + 38 39) &= -1 + (1 2 - 2 3) + (3 4 - 4 5) + + (37 38 - 38 39) + 39 &= -1 + 2(-2) + 4(-2) + + 38(-2) + 39 &= -1 - 2 2 19 202 + 39 &= -1 - 760 + 39 &= -722. alignedThus, |A-B| = 722.