1的平方一直加到n的平方等于多少 1平方加到n平方的推导是？
\u4ece1\u7684\u5e73\u65b9\u4e00\u76f4\u52a0\u5230N\u7684\u5e73\u65b9\u7b49\u4e8e\u591a\u5c11\u4ece1\u7684\u5e73\u65b9\u4e00\u76f4\u52a0\u5230N\u7684\u5e73\u65b9\u7684\u548c\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a
1^2 + 2^2 + 3^2 + ... + N^2
\u8fd9\u4e2a\u548c\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\uff1a
N(N+1)(2N+1)/6
\u6240\u4ee5\uff0c\u4ece1\u7684\u5e73\u65b9\u4e00\u76f4\u52a0\u5230N\u7684\u5e73\u65b9\u7684\u548c\u7b49\u4e8eN(N+1)(2N+1)/6\u3002
1\u7684\u5e73\u65b9\u52a0\u5230n\u7684\u5e73\u65b9\u7684\u63a8\u5bfc\u516c\u5f0f\u5982\u4e0b\uff1a1²+2²+3²+\u2026\u2026+n²=n(n+1)(2n+1)/6\u3002
\u6839\u636e\u7acb\u65b9\u5dee\u516c\u5f0f(a+1)³-a³=3a²+3a+1\u53ef\u5f97\uff0ca=1\u65f6\uff1a2³-1³=3\u00d7bai1²+3\u00d71+1\uff0ca=n\u65f6\uff1a(n+1)³-n³=3\u00d7n²+3\u00d7n+1\uff0c\u5c06\u591a\u4e2a\u7b49\u5f0f\u76f8\u52a0\uff0c\u65e2\u67092(n+1)³-3n(1+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)\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u7acb\u65b9\u5dee\u516c\u5f0f\u4e0e\u7acb\u65b9\u548c\u516c\u5f0f\u4e00\u8d77\u5408\u79f0\u4e3a\u5b8c\u5168\u7acb\u65b9\u516c\u5f0f\u3002\u7acb\u65b9\u5dee\u516c\u5f0f\u6307\u7684\u662f\uff1a\u6570\u7684\u5e73\u65b9\u548c\u52a0\u4e0a\u4e24\u6570\u7684\u79ef\u518d\u4e58\u4ee5\u4e24\u6570\u7684\u5dee\uff0c\u6240\u5f97\u5230\u7684\u79ef\u5c31\u7b49\u4e8e\u4e24\u6570\u7684\u7acb\u65b9\u5dee\u3002
\u7acb\u65b9\u5dee\u516c\u5f0f\u7684\u8bc1\u660e\u5982\u4e0b\uff1a
a3-b3=a3-b3+a2b-a2b
=a2(a-b)+b(a2-b2)
=a2(a-b)+b(a+b)(a-b)
=[a2+b(a+b)](a-b)
=(a-b)(a2+ab+b2)
这个公式不要求证明，记住就可以了。
公式的延伸
供参考，请笑纳。
1²+2²+²+···+（n-1）²+n²
=n(n+1)(2n+1)/6
绛旓細=n^2+(n-1)^2+n^2-n =2*n^2+(n-1)^2-n 鍚勭瓑寮忓叏鐩稿姞灏卞緱鍒板挴銆
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