根据下面一组等式.分别计算S1,S2,S3,S4,S5,S6如上,试用不完全归纳法猜测可得根据下面一组等式：S1=1,S2=2+3=5,S3=4+5+6=15,S4=7+8+9+10=34,S5=11+12+13+14+15=65,S6=16+17+18+19+20+21=111.分别计算S1,S2,S3,S4,S5,S6如上,试用不完全归纳法猜测可得S1+S3+S5+...+S(2n-1)的结果,并用数学归纳法证明,

S1 = 1 = 1^4
S1+S3 = 16 = 2^4
S1+S3+S5 = 81 = 3^4
S1+S3+S5+...+S(2n-1) = n^4
当n=1时,成立
设当n=k(>1)时 S1+S3+S5+...+S(2k-1) = k^4
当n= k+1时 S1+S3+S5+...+S(2k-1)+S(2k+1) = k^4 + S(2k+1)
= k^4 + (1+2k)*(2k)/2 *(2k+1) + (1+2k+1)/2 * (2k+1)
= k^4 + 4k^3 + 6k^2 + 4k + 1
= (k+1)^4
得证