一的平方加二的平方加三的平方·····一直加到n的平方等于多少 1的平方加2的平方一直加到n的平方等于多少
1\u5e73\u65b9\u52a02\u5e73\u65b9\u52a03\u5e73\u65b9\u4e00\u76f4\u52a0\u5230n\u5e73\u65b9\u7b49\u4e8e\u591a\u5c111²+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
=n(n+1)(2n+1)/6
(n+1)^3-n^3=3n^2+3n+1
n^3-(n-1)^3=3(n-1)^2+3(n-1)+1
……
2^3-1^3=3*1^2+3*1+1
全都加起来,左边中间可以抵消掉
(n+1)^3-1=3*[n^2+(n-1)^2+(n-2)^2+……+2^2+1^2]+3*[n+(n-1)+……+3+2+1]+n*1
而(n+1)^3-1=(n+1-1)[(n+1)^2+(n+1)+1]
=n(n^2+3n+3)
n+(n-1)+……+3+2+1=n(n+1)/2
所以n^2+(n-1)^2+(n-2)^2+……+2^2+1^2={[(n+1)^3-1]-3*[n+(n-1)+……+3+2+1]-n*1}/3
=[n(n^2+3n+3)-3n(n+1)/2-n]/3
=[n(n^2+3n+3-3n/2-3/2-1)]/3
=n(2n^2+3n+1)/6
=n(n+1)(2n+1)/6
∵(n＋1)³＝n³＋3n²＋3n＋1
∴ 1³＝1³
2³＝1³＋3×1²＋3×1＋1
3³＝2³＋3×2²＋3×2＋1
……
n³＝(n－1)³＋3(n－1)²＋3(n－1)＋1
(n＋1)³＝n³＋3n²＋3n＋1
上述各式相加得：(n＋1)³＝1³＋3×(1²＋2²＋3²＋……＋n²)＋3×(1＋2＋3＋……＋n)＋n
∴3×(1²＋2²＋3²＋……＋n²)＝ (n＋1)³－n－1－3×(1＋2＋3＋……＋n)
＝n³＋3n²＋2n－3/2×n(n＋1)＝1/2×(2n³＋3n²＋n)＝n(2n＋1)(n＋1)/2
∴1²＋2²＋3²＋……＋n²＝n(2n＋1)(n＋1)/6
1^2+2^2+3^2+…+n^2=n(n+1)(2n+1)/6 这是公式
1^2+2^2+3^2+…+n^2=n(n+1)(2n+1)/6
绛旓細=n(n+1)(2n+1)/6
绛旓細1锛4锛9锛嬧︼紜x2锛嬶紙x锛1锛2锛漻(x+1)(2x+1)/6锛嬶紙x锛1锛2 锛濓紙x锛1锛塠2锛坸2锛夛紜x锛6锛坸锛1锛塢/6 锛濓紙x锛1锛塠2锛坸2锛夛紜7x锛6]/6 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...
绛旓細=338350
绛旓細鍥炵瓟锛氭湁鍏垎涔嬩竴n(n+1)(2n+1)
绛旓細鍏紡锛1²+2²+3²+.+N²=n锛坣+1锛(2n+1)/6 1鐨勫钩鏂瑰姞鍒100鐨勫钩鏂 =100脳101脳201锛6=338350
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1²+2²+3²+鈥︹+n²=n(n+1)(2n+1)/6銆傚彲浠ョ敤(n+1)³-n³=3n²+3n+1绱姞寰楀埌銆傝瘉鏄庤繃绋嬶細鏍规嵁绔嬫柟宸叕寮(a+1)³-a³=3a²+3a+1锛屽垯鏈夛細a=1鏃讹細2³-1³=3脳1²+3脳1+1 a=2鏃讹細3³-2&#...
绛旓細4^3-3^3=2*4^2+3^2-4 ...n^3-(n-1)^3=2*n^2+(n-1)^2-n 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+...
绛旓細瑕佸垪鍑1鐨勫钩鏂瑰姞2鐨勫钩鏂瑰姞3鐨勫钩鏂涓鐩村姞鍒10鐨勫钩鏂圭殑椤癸紝鎴戜滑鍙互浣跨敤鏁板绗﹀彿鏉ヨ〃绀恒傝繖涓簭鍒楀彲浠ヨ〃绀轰负锛1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10²绠鍖栦竴涓嬶紝鎴戜滑鍙互鍐欐垚锛氣垜(n²), n浠1鍒10...
绛旓細鍥炵瓟锛氫唬鍏(n+1)(2n+1)/6
