Simplify (220+320)(221+321)(222+322)⋯(2210+3210)+2204832048.
My solution:
(220+320)(221+321)(222+322)⋯(2210+3210)+2204832048
=(2+3)(22+32)(24+34)⋯(21024+31024)+2204832048
=(3+2)(32+22)(34+24)⋯(31024+21024)+2204832048
=(3+2)(3−2)⏟(32+22)(34+24)⋯(31024+21024)+22048(3−2)(32048)
=(32−22)(32+22)⏟(34+24)⋯(31024+21024)+22048(3−2)(32048)
=(34−24)(34+24)⏟⋯(31024+21024)+22048(3−2)(32048)
=(38−28)⋯(31024+21024)+22048(3−2)(32048)
=(31024−21024)(31024+21024)⏟+22048(3−2)(32048)
=(32048−22048)+22048(1)(32048)
=(32048(32048)
=1
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