一的平方加二的平方加三的平方一直加到二零1七的平方除以一加二加三 1平方加2平方加3平方一直加到n平方等于多少
\u6c42\u4e00\u7684\u5e73\u65b9\u52a0\u4e8c\u5e73\u65b9\u52a0\u4e09\u5e73\u65b9\u52a0\u56db\u5e73\u65b9\u4e00\u76f4\u52a0\u5230\u4e00\u767e\u7684\u5e73\u65b9\u9664\u4ee5\u4e03\u7684\u4f59\u65701²+2²+3²+\u3002\u3002\u3002+n²=n\uff08n+1)\uff082n+1\uff09/6
\u628an=100\u4ee3\u5165\u4e0a\u5f0f\uff0c\u5c31\u53ef\u4ee5\u7b97\u51fa\u6765\u4e86\uff0c\u4f59\u6570\u662f5
1²+2²+3²+\u2026\u2026+n²=n(n+1)(2n+1)/6\u3002\u53ef\u4ee5\u7528(n+1)³-n³=3n²+3n+1\u7d2f\u52a0\u5f97\u5230\u3002
\u8bc1\u660e\u8fc7\u7a0b\uff1a
\u6839\u636e\u7acb\u65b9\u5dee\u516c\u5f0f(a+1)³-a³=3a²+3a+1\uff0c\u5219\u6709\uff1a
a=1\u65f6\uff1a2³-1³=3\u00d71²+3\u00d71+1
a=2\u65f6\uff1a3³-2³=3\u00d72²+3\u00d72+1
a=3\u65f6\uff1a4³-3³=3\u00d73²+3\u00d73+1
a=4\u65f6\uff1a5³-4³=3\u00d74²+3\u00d74+1.\u00b7\u00b7
a=n\u65f6\uff1a\uff08n+1\uff09³-n³=3\u00d7n²+3\u00d7n+1
\u7b49\u5f0f\u4e24\u8fb9\u76f8\u52a0\uff1a
\uff08n+1)³-1=3\uff081²+2²+3²+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n²\uff09+3\uff081+2+3+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n\uff09+\uff081+1+1+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+1\uff09
3\uff081²+2²+3²+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n²\uff09=\uff08n+1\uff09³-1-3\uff081+2+3+.+n\uff09-\uff081+1+1+.+1\uff09
3\uff081²+2²+3²+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n²\uff09=\uff08n+1\uff09³-1-3\uff081+n\uff09\u00d7n\u00f72-n
6\uff081²+2²+3²+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n²\uff09=2\uff08n+1)³-3n\uff081+n)-2(n+1)=(n+1)[2(n+1)²-3n-2]
=(n+1)[2(n+1)-1][(n+1)-1]=n(n+1)(2n+1)
\u6240\u4ee51²+2²+\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7+n²=n\uff08n+1)\uff082n+1\uff09/6\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u7acb\u65b9\u5dee\u516c\u5f0f\u4e0e\u7acb\u65b9\u548c\u516c\u5f0f\u7edf\u79f0\u4e3a\u7acb\u65b9\u516c\u5f0f\uff0c\u4e24\u8005\u57fa\u672c\u63cf\u8ff0\u5982\u4e0b:
1\u3001\u7acb\u65b9\u548c\u516c\u5f0f\uff0c\u5373\u4e24\u6570\u7acb\u65b9\u548c\u7b49\u4e8e\u8fd9\u4e24\u6570\u7684\u548c\u4e0e\u8fd9\u4e24\u6570\u5e73\u65b9\u548c\u4e0e\u8fd9\u4e24\u6570\u79ef\u7684\u5dee\u7684\u79ef\u3002\u4e5f\u53ef\u4ee5\u8bf4\u4e24\u6570\u7acb\u65b9\u548c\u7b49\u4e8e\u8fd9\u4e24\u6570\u79ef\u4e0e\u8fd9\u4e24\u6570\u5dee\u7684\u4e0d\u5b8c\u5168\u5e73\u65b9\u7684\u79ef\u3002
2\u3001\u7acb\u65b9\u5dee\u516c\u5f0f\uff0c\u5373\u4e24\u6570\u7acb\u65b9\u5dee\u7b49\u4e8e\u8fd9\u4e24\u6570\u5dee\u4e0e\u8fd9\u4e24\u6570\u5e73\u65b9\u548c\u4e0e\u8fd9\u4e24\u6570\u79ef\u7684\u548c\u7684\u79ef\u3002\u4e5f\u53ef\u4ee5\u8bf4\uff0c\u4e24\u6570\u7acb\u65b9\u5dee\u7b49\u4e8e\u4e24\u6570\u5dee\u4e0e\u8fd9\u4e24\u6570\u548c\u7684\u4e0d\u5b8c\u5168\u5e73\u65b9\u7684\u79ef \u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1_\u7acb\u65b9\u5dee\u516c\u5f0f
1+2+3+.....+n=[n(n+1)]/2
就可以求出来了。
一的平方加二的平方加三的平方一直加到二零1七的平方除以一加二加三一直加到2017
=[(2017x2018x4035)/6]/[2017x2018)/2]
=4035x2/6
=1345。
（1+2+3+……+2017²）÷（1+2+3+……+2017）
=[2017×（2017+1）×（2017×2+1）÷6]÷[2017×（2017+1）÷2]
=1345
绛旓細锛濓紙x锛1锛夛紙2x锛3锛夛紙x锛2锛/6 锛濓紙x锛1锛塠锛坸锛1锛夛紜1][2锛坸锛1锛+1]/6 涔熸弧瓒冲叕寮 4銆佺患涓婃墍杩,骞虫柟鍜屽叕寮1^2+2^2+3^2+鈥+n^2=n(n+1)(2n+1)/6鎴愮珛,寰楄瘉.璇佹硶浜岋紙鍒╃敤鎭掔瓑寮(n+1)^3=n^3+3n^2+3n+1锛夛細(n+1)^3-n^3=3n^2+3n+1,n^3-(n-1)^3=3...
绛旓細=n(n+1)(2n+1)/6
绛旓細鍚勭瓑寮忓叏鐩稿姞 n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n)n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+...+n)n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...
绛旓細杩欏叾瀹炲氨鏄竴閬撳簭鍒楅锛岀孩濮愬洖绛旂殑鏄纭殑锛宻=1^2+2^2+...+100^2 =100[100+1][2*100+1]/6 =338350 S锛坣)=n(n+1)(2n+1)/6
绛旓細鍒╃敤鍏紡锛1^2+2^2+3^3+...+n^2=[n(n+1)(2n+1)]/6 1+2+3+...+n=[n(n+1)]/2 灏卞彲浠ユ眰鍑烘潵浜嗐涓鐨勫钩鏂瑰姞浜岀殑骞虫柟鍔犱笁鐨勫钩鏂涓鐩村姞鍒颁簩闆1涓冪殑骞虫柟闄や互涓鍔浜屽姞涓変竴鐩村姞鍒2017 =[(2017x2018x4035)/6]/[2017x2018)/2]=4035x2/6 =1345銆
绛旓細瑙ｏ細鍏紡锛1²+2²+3²+...+n²=1/6 n(n+1)(2n+1)鎵浠 鍙杗=100锛屽緱 鍘熷紡=1/6 脳100脳锛100+1锛壝楋紙2脳100+1锛=338350
绛旓細鍥炵瓟锛氫唬鍏(n+1)(2n+1)/6
绛旓細鍥炵瓟锛氭湁鍏垎涔嬩竴n(n+1)(2n+1)
绛旓細鎵浠1鐨勫钩鏂+2鐨勫钩鏂+3鐨勫钩鏂+鈥+n鐨勫钩鏂=n(n+1)(2n+1)/6 绠浠 鏁板褰掔撼娉曪紙Mathematical Induction, MI锛夋槸涓绉嶆暟瀛﹁瘉鏄庢柟娉曪紝閫氬父琚敤浜庤瘉鏄庢煇涓粰瀹氬懡棰樺湪鏁翠釜锛堟垨鑰呭眬閮級鑷劧鏁拌寖鍥村唴鎴愮珛銆傞櫎浜嗚嚜鐒舵暟浠ュ锛屽箍涔変笂鐨勬暟瀛﹀綊绾虫硶涔熷彲浠ョ敤浜庤瘉鏄庝竴鑸壇鍩虹粨鏋勶紝渚嬪锛氶泦鍚堣涓殑鏍戙傝繖绉嶅箍涔夌殑鏁板...
绛旓細...19* 20^2= 20^3 - 20^2 鍘熷紡=2^3 - 2^2+3^3 - 3^2+ 4^3 - 4^2銆傘傘+20^3 - 20^2 =锛1+2^3 +3^3 + 4^3 銆傘傘+20^3 锛-锛1+2^2 +3^2 + 4^2 銆傘傘+20^2锛=銆愶紙20*21锛/2銆慯2-20*锛20+1锛*锛40+1锛/6 =210^2-2870 =41230 ...
