5^2－3^2=8×2，9^2－7^2=8×4,15^2－5^2=8×25 .用文字或字母表示上述算式的规律,并证明这个规律的正确性 老师在黑板上写出三个算式：5^2－3^2=8×2，9^2—7...
\u8001\u5e08\u5728\u9ed1\u677f\u4e0a\u5199\u6709\u4e09\u4e2a\u7b97\u5f0f:5^2-3^2=8*2,9^2-7^2=8*4,15^2-3^2=8*27,\u674e\u660e\u63a5\u7740\u53c8\u5199\u4e24\u89e3\u7b54\uff1a\uff081\uff0919^2-15^2=8x17\uff1b 21^2-15^2=8x27\u3002
\uff082\uff09 \u4e0a\u8ff0\u7b97\u5f0f\u7684\u89c4\u5f8b\u662f \u4e24\u4e2a\u6b63\u5947\u6570\u7684\u5e73\u65b9\u5dee\u4e00\u5b9a\u662f8\u7684\u6574\u6570\u500d\u3002
\uff083\uff09 \u8bc1\u660e\uff1a(2n+1)^2-(2k+1)^2
=4n^2+4n+1-4k^2-4k-1
=4(n^2-k^2)-4(n+k)
=4(n+k)(n-k)+4(n+k)
=4(n+k)(n-k+1)
\u56e0\u4e3a n+k\u548cn-k+1\u5947\u5076\u6027\u76f8\u53cd\uff0c\u5373n+k\u548cn-k+1 \u4e00\u5b9a\u6709\u4e00\u4e2a\u662f\u5076\u6570\uff0c
\u6240\u4ee5 4(n+k)(n-k+1)\u4e00\u5b9a\u662f8\u7684\u500d\u6570\u3002
\u6ce8\u610f\uff1a\u8fd9\u4e2a\u89c4\u5f8b\u53ea\u9002\u7528\u4e8e\u5947\u6570\u7684\u5e73\u65b9\u5dee\u3002
5^2=25 3^2=9 8*2=16 15^2=225 3^2=9 8*27=216
9^2=81 7^2=49 8*4=32 ()^2-----()^2=49 8*54=432
\u89e3\u4e0d\u51fa \u6700\u597d\u770b\u4e00\u4e0b\u662f\u4e0d\u662f\u5199\u9519\u4e86= =
证明：
a+b=4n a-b= 2k 所以 4n*2=8nk
据此得证
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呃，看不懂……= =|||||||||||
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