n平方+(n+3)平方+(n+5)平方+(n+6)平方(=n+1)平方+(n+2)平方+(n+4)平方+(n+7)平方

∵左边=4×n平方+6n+10n+12n+9+25+36=4×n平方+28n+70
右边=4×n平方+2n+4n+8n+14n+1+4+16+49=4×n平方+28n+70
∴左边=右边，即n平方+(n+3)平方+(n+5)平方+(n+6)平方(=n+1)平方+(n+2)平方+(n+4)平方+(n+7)平方

自己展开就行了啊，I服了YOU！

(n+1)^2+(n+2)^2+(n+4)^2+(n+7)^2

(n+7)^2-(n+6)^2=(n+7-n-6)(n+7+n+6)=2n+13
(n+4)^2-(n+5)^2=(n+4-n-5)(n+4+n+5)=-2n-9
(n+2)^2-(n+3)^2=(n+2-n-3)(n+2+n+3)=-2n-5
(n+1)^2-n^2=(n+1-n)(n+1+n)=2n+1

(n+1)^2+(n+2)^2+(n+4)^2+(n+7)^2-[n^2+(n+3)^2+(n+5)^2+(n+6)^2]
=2n+13-2n-9-2n-5+2n+1
=0

所以
(n+1)^2+(n+2)^2+(n+4)^2+(n+7)^2=n^2+(n+3)^2+(n+5)^2+(n+6)^2

n²+(n+3)²+(n+5)²+(n+6)²
=[(n+1)-1]²+[(n+2)+1]²+[(n+4)+1]²+[(n+7)-1]²
=[(n+1)²-2(n+1)+1]+[(n+2)²+2(n+2)+1]+[(n+4)²+2(n+4)+1]+[(n+7)²-2(n+7)+1]
=[(n+1)²+(n+2)²+(n+4)²+(n+7)²]+2[-(n+1)+(n+2)+(n+4)-(n+7)]+4
=[(n+1)²+(n+2)²+(n+4)²+(n+7)²]+2*(-1+2+4-7)+4
=[(n+1)²+(n+2)&s