类比推理2^2-1^2=2*1+13^2-2^2=2*2+14^2-3^2=2*3+1(n+1)^2-n^2=2n+1(n+1)^2-1^2=2(1+2+3.+n)+n即1+2+3..+n=n(n+1)/2(1)类比上述求法,请求出1^2+2^2+3^2+...+n^2(2)根据上述结论试求1^2+3^2+5^2+...+99^2的值
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![类比推理2^2-1^2=2*1+13^2-2^2=2*2+14^2-3^2=2*3+1(n+1)^2-n^2=2n+1(n+1)^2-1^2=2(1+2+3.+n)+n即1+2+3..+n=n(n+1)/2(1)类比上述求法,请求出1^2+2^2+3^2+...+n^2(2)根据上述结论试求1^2+3^2+5^2+...+99^2的值](/uploads/image/z/9616161-57-1.jpg?t=%E7%B1%BB%E6%AF%94%E6%8E%A8%E7%90%862%5E2-1%5E2%3D2%2A1%2B13%5E2-2%5E2%3D2%2A2%2B14%5E2-3%5E2%3D2%2A3%2B1%28n%2B1%29%5E2-n%5E2%3D2n%2B1%28n%2B1%29%5E2-1%5E2%3D2%281%2B2%2B3.%2Bn%29%2Bn%E5%8D%B31%2B2%2B3..%2Bn%3Dn%28n%2B1%29%2F2%EF%BC%881%EF%BC%89%E7%B1%BB%E6%AF%94%E4%B8%8A%E8%BF%B0%E6%B1%82%E6%B3%95%2C%E8%AF%B7%E6%B1%82%E5%87%BA1%5E2%2B2%5E2%2B3%5E2%2B...%2Bn%5E2%EF%BC%882%EF%BC%89%E6%A0%B9%E6%8D%AE%E4%B8%8A%E8%BF%B0%E7%BB%93%E8%AE%BA%E8%AF%95%E6%B1%821%5E2%2B3%5E2%2B5%5E2%2B...%2B99%5E2%E7%9A%84%E5%80%BC)
类比推理2^2-1^2=2*1+13^2-2^2=2*2+14^2-3^2=2*3+1(n+1)^2-n^2=2n+1(n+1)^2-1^2=2(1+2+3.+n)+n即1+2+3..+n=n(n+1)/2(1)类比上述求法,请求出1^2+2^2+3^2+...+n^2(2)根据上述结论试求1^2+3^2+5^2+...+99^2的值
类比推理
2^2-1^2=2*1+1
3^2-2^2=2*2+1
4^2-3^2=2*3+1
(n+1)^2-n^2=2n+1
(n+1)^2-1^2=2(1+2+3.+n)+n
即1+2+3..+n=n(n+1)/2
(1)类比上述求法,请求出1^2+2^2+3^2+...+n^2
(2)根据上述结论试求1^2+3^2+5^2+...+99^2的值
类比推理2^2-1^2=2*1+13^2-2^2=2*2+14^2-3^2=2*3+1(n+1)^2-n^2=2n+1(n+1)^2-1^2=2(1+2+3.+n)+n即1+2+3..+n=n(n+1)/2(1)类比上述求法,请求出1^2+2^2+3^2+...+n^2(2)根据上述结论试求1^2+3^2+5^2+...+99^2的值
(n+1)^3 - n^3 = (n^3 + 3n^2 + 3n + 1) - n^3 = 3*n^2 + 3n + 1
利用上面这个式子有:
2^3 - 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
5^3 - 4^3 = 3*4^2 + 3*4 + 1
……
(n+1)^3 - n^3 = 3*n^2 + 3n + 1
把上述各等式左右分别相加 得到:
(n+1)^3 - 1^3 = 3*(1^2+2^2+3^2+……+n^2) + 3*(1+2+3+……+n) + n*1
n^3 + 3n^2 + 3n + 1 - 1 = 3*(1^2+2^2+3^2+……+n^2) + 3*n(n+1)/2 + n
继续整理(属于纯粹的数学运算了),最后
1^2 + 2^2 + 3^2 + …… + n^2 = n(n+1)(2n+1)/6
第二道题带入99就行了,自己算吧