1平方加到n平方简算过程及证明方法
因为（k+1）^3=k^3+3k^2+3k+1.1k^3=(k-1)^3+3(k-1)^2+3(k-1)+1.2
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2^3=1^3+3*1^2+3*1+1.k
k式相加：(k+1)^3-1=3(k^2+.+1)+3(k+k-1+.+1)+k
所以3（k^2+...+1)
=(k+1)[(k+1)^2-1-k-(3k(k+1)/2)]
=k(k+1)(2k+1)
故1^2+2^2+3^2+...+N^2=N(N+1)(2N+1)/6
(n+1)^3-n^3=3n^2+3n+12^3-1^3=3*1^2+3*1+1
3^3-2^3=3*2^2+3*2+1
4^3-3^3=3*3^2+3*3+1
.(n+1)^3-n^3=3n^2+3n+1
以上各式相加,可得:(n+1)^3-1^3=3(
1^2+2^2+3^2+...+n^2)+3(1+2+3+4+5+6+.+n)+n
即n^3+3n^2+3n=3(1^2+2^2+3^2+...+n^2)+3n(n+1)/2+n
整理即可得1^2+2^2+3^2+...+N^2=N(N+1)(2N+1)/6
绛旓細2^2+4^2+...+2004^2 =(2*1)^2+(2*2)^2+...+(2*1002)^2 =2^2*(1^2+2^2+...+1002^2)=4*1002*(1002+1)*(2*1002+1)/6 =1343358020 鍏紡锛 1^2+2^2+...n^2=n(n+1)(2n+1)/6
绛旓細1.1鐨勫钩鏂瑰姞鍒10鐨勫钩鏂逛釜浣嶆暟鏄5,1鍒1990鍏辨湁199缁勶紝鎵浠ュ姞鍒1990涓綅鏁版槸5锛屽啀鍔犲洓涓彲鐭ヤ釜浣嶆暟鏄5.2.鍘熷紡=锛1+1/2+1/3+1/4锛夛紙1/2+1/3+1/4+1/5锛-锛1+1/2+1/3+1/4锛夛紙1/2+1/3+1/4锛-1/5锛1/2+1/3+1/4锛=锛1+1/2+1/3+1/4锛*1/5-锛1/2+1/3+1/...
绛旓細..+2011)+(1²+2²+3²+4²+...+2011²)]甯﹀叆1+2²+3²+...+n²=n(n+1)(2n+1)/6 = ½[(1006*2011)+(2011*2012*4023)/6]=½(2023066+2712931506)=1357477286 鏍囧噯鏍煎紡濡備笅锛氬弬鑰冭祫鏂欙細骞虫柟鍜 ...
绛旓細1-2+3-4+5-6+鈥︹︹-100+101 =锛1+2锛夛紙1-2锛+锛3+4锛夛紙3-4锛+鈥︹+锛99+100锛夛紙99-100锛+101 =-锛1+2锛-锛3+4锛-鈥︹︹旓紙99+100锛+101 =锛-1锛壝楋紙1+2+3+鈥︹+100锛+101 =101-5050 =10201-5050 =5151
绛旓細-2008*20092009+2009*20082008=-2008*2009*10001+2009*2008*10001 =0 =1/4-1/5+1/5-1/6鈥︹+1/11-1/12=1/4-1/12=1/6 浠1+5+5鐨勫钩鏂+5鐨勭珛鏂+...+5鐨25娆℃柟=A 閭ｄ箞5A=5+5鐨勫钩鏂+5鐨勭珛鏂+...+5鐨26娆℃柟 5A-A=5鐨26娆℃柟-1=4A A=锛5鐨26娆℃柟-1锛/4 ...
绛旓細1-25骞虫柟锛1²=1 銆2²=4銆 3²=9銆 4²=16銆 5²=25銆 6²=36銆 7²=49銆 8²=64銆 9²=81銆 10²=100銆 11=121 銆12²=144銆 13²=169銆 14²=196銆 15²=225銆 16²=256銆 17²...
绛旓細1^2+2^2+3^2...+n^2=1/6*n(n+1)(2n+1)7^2+8^2+9^2+10^2+11^2+12^2 =[1^2+2^2+3^2...+11^2+12^2]-[1^2+2^2+3^2...+6^2]=1/6*12*13*25-1/6*6*7*13=559 杩欐牱鍙互璁＄畻n^2+(n+1)^2+...m^2浜 鍏充簬1^2+2^2+3^2...+n^2=1/6*n(n+...
绛旓細89鍜98蹇呴夊叾涓,閫夊嚭1涓.濡傛灉鎴戜滑鍙変袱涓腑鐨勫皬鏁拌繖鏍峰皢浼氶夊嚭9+8+7+6+5+4+3+2+1=45涓.鍐鍔犱笂11~99杩9涓暟灏辨槸54涓.6銆佺敤閫氬垎绠楁硶鐭ラ亾锛1/[n(n+1)] = 1/n - 1/(n+1)鎵浠ワ細1*2鍒嗕箣1+2*3鍒嗕箣1+3*4鍒嗕箣1+鈥︹+199*200鍒嗕箣1 =(1-1/2) + (1/2-1/3) + (...
绛旓細1^2-2^2+3^3-4^2+鈥︹+2003^2-2004^2+2005^2 =1^2+(3^2-2^2)+(5^2-4^2)+...+(2005^2-2004^2)=1^2+(3+2)(3-2)+(5+4)(5-4)+...+(2005+2004)(2005-2004)=1^2+5+9+...+4009 =1^2+(5+4009)脳[(4009-5)梅4锛1]梅2 =1+2011014 =2011015 ...
绛旓細鍒╃敤骞虫柟鍜鍏紡鍚э細n(n+1)(2n+1)/6 30*(30+11)*(2*30+11)/6=5*41*71
