Question

1²+3²+5²+…+25²=?

Answer 1

1^2+2^2+3^2+4^2+5^2+6^2+……n^2=n(n+1)(2n+1)/6 (这是公式，课本上有的)

则
1^2+2^2+3^2+4^2+5^2+6^2+……+(2n)^2=2n(2n+1)(2*2n+1)/6=n(2n+1)(4n+1)/3

即
[1^2+3^2+5^2+......+(2n-1)^2]+[2^2+4^2+6^2+.....+(2n)^2]=n(2n+1)(4n+1)/3
即
[1^2+3^2+5^2+......+(2n-1)^2]+4(1^2+2^2+3^2+......+n^2)=n(2n+1)(4n+1)/3将已知等式1^2+2^2+3^2+4^2+5^2+6^2+……n^2=n(n+1)(2n+1)/6代入

得
[1^2+3^2+5^2+......+(2n-1)^2]+2n(n+1)(2n+1)/3=n(2n+1)(4n+1)/3

所以
1^2+3^2+5^2+......+(2n1)^2 = n(2n+1)(4n+1)/3-2n(n+1)(2n+1)/3
=n(2n+1)(2n-1)/3
将n=25代入得
1²+3²+5²+…+25²=25*（2*25+1）（2*25-1）/3=20825

Answer 2

1^2+2^2+……+n^2=(n^3+3n^2+3n)/3-n(n+1)/2-n/3=n(n+1)(2n+1)/6

所以1²+2²+3²+4²+5²+……+24²+25²=5525