乘式求和简算原理 (一) 1×2+2×3+···+5×6 因为: 1×2=(1×2×3-0×1×2)÷(1×3) 2×3=(2×3×4-1×2×3)÷3(除的这个3来源于两个单项式中的最大因数-最小因数的差,也就是÷实际扩大的倍数,还原2×3的值。也可理解为:公差×(单项因素个数+1)) ················· 5×6=(5×6×7-4×5×6)÷3 所以: 1×2+2×3+····+5×6 =(5×6×7-0×1×2)÷3 (除最大最小乘式,其余全部正负抵销) =210÷3 =70
(二) 1×3+3×5+5×7+······+97×99 因为: 1×3=(1×3×5-(-1)×1×3)÷(5-(-1)) (最大-最小) =(15+3)÷6 =3 3×5=(3×5×7-1×3×5)÷6 5×7=(5×7×9-7×5×3)÷6 ················· 97×99=(97×99×101-99×97×95)÷6 所以: 1×3+3×5+5×7+····97×99 =(97×99×101-(-1)×1×3)÷6 (除最大最小乘式,其余全部正负抵销) =161651 (三) 5×6+6×7+····+9×10 因为: 5×6=(5×6×7-4×5×6)÷3 6×7=(6×7×8-5×6×7)÷3 ················· 9×10=(9×10×11-8×9×10)÷3 所以: 5×6+6×7+····+9×10 =(9×10×11-4×5×6)÷3 =(990-120)÷3 =870÷3 =290 (四) 3×6×9+6×9×12+·······+12×15×18 因为: 3×6×9=(3×6×9×12- 0×3×6×9)÷(3×4) (公差×(单项因素个数+1)) 6×9×12=(6×9×12×15-3×6×9×12)÷12 ················ 12×15×18=(12×15×18×21-9×12×15×18)÷12 所: 3×6×9+6×9×12+·······+12×15×18 =(12×15×18×21-0×3×6×9)÷12 (除最大最小乘式外,其余全部正负抵销) =5670 (五) 5×10×15×20+10×15×20×25+15×20×25×30 因为: 5×10×15×20=(5×10×15×20×25-0×5×10×15×20)÷25 10×15×20×25=(10×15×20×25×30-5×10×15×20×25)÷25 15×20×25×30= (15×20×25×30×35- 10×15×20×25×30)÷25 所以: 5×10×15×20+10×15×20×25+15×20×25×30 =(15×20×25×30×35--0×5×10×15×20)÷25 =(7875000-0)÷25 =315000
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