Chapter 41 second segmental surgery

The arithmetic series problem developed into a high-order arithmetic series summation problem in the Song and Yuan dynasties.The pioneer of this subject was Shen Kuo, a great scientist in the Northern Song Dynasty.Shen Kuo studied the volume formula of the three-dimensional volume of Chu Tong in "Nine Chapters of Arithmetic" (see Section 4), and thought it was quite complete, but there was no method for calculating the gap product.Gap product is the product with gaps, such as stacking chess pieces, altars, pots, etc., as shown in Figure 34. Although it has the shape of a chutong, but because of the gaps, if you use the chutong technique to calculate the product, the value On the small side.Shen Kuo proposed the gap accumulation technique.Assuming that the upper base of the gap product has width a, length b, lower bottom width a, length b, and height n floors, and aabb=n-1, the gap product technique proposed by Shen Kuo is: S=ab+(a+1)(b+ 1)+(a+2)(b+2)+...+ab=n[(2a+a)b+(2a+a)b+(aa)]/6

That is, the number of objects in the cuboid volume is n/6×(aa) more than the volume of the cuboid.Gap product is actually a second-order arithmetic series summation problem: b(a+1)(b+1)+2)(b+2)+3)(b+3)+4)(b+4)... +b+1+b+3+b+5+b+7… Yang Hui of the Southern Song Dynasty's "Detailed Explanation of the Nine Chapters Algorithm" compares the three-dimensionality of "Nine Chapters" with various fruit stacks.Among them, the child-shaped fruit pile is the same as Shen Kuo's gap accumulation technique.The quadrature formula of Sijiao Duo (Bi Lei Square Cone, Yang Ma) is: S=1+2+3+...+n=n(n+1)(n+1/2)/3

The quadrature formula of the square pile (compared to the square pavilion) is: S=a+(a+1)+(a+2)+...+(b-1)+b=n〔a+b+ab+1/2 (ba)〕/3 The quadrature formula of triangular stacks (compared to similar turtles) is: S=1+3+6+10+...+1/2n(n+1)=n(n+1)(n+2)/6 It is not difficult to see that this is a second-order arithmetic series summation problem.At the same time, it can be seen that in Shen Kuo's gap accumulation technique, let a=b=1, a=b=n, which is Yang Hui's four-corner stack formula; let a=b, a=b, is Yang Hui's square stack formula ; Let a=1, b=2, a=n, b=n+1, it becomes the sum of two triangular stacks 1·2+2·3+3·4+...+n(n+1)=n (n+1)(n+2)/3.

Divide both ends by 2 to get Yang Hui's triangular stack formula.