Problem:
The expressions A=1×2+3×4+5×6+⋯+37×38+39 and B= 1+2×3+4×5+⋯+36×37+38×39 are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers A and B.
Solution:
Observe
B−A=(1−39)+(3×2−1×2)+(5×4−3×4)+(7×6−5×6)+⋯+(39×38−37×38)=−38+(2×2)+(2×4)+(2×6)+⋯+(2×38)=−38+4×(1+2+3+⋯+19)=−38+4×219⋅20=722.
The problems on this page are the property of the MAA's American Mathematics Competitions