44……488……89
=44.488.88+1 (n个4,n个8)
=4*11.1*10^n+8*11.1+1 (n个1,n个1)
=(4/9)(99.9)*10^n+(8/9)(99.9)+1 (n个9,n个9)
=(4/9)(10^n-1)*10^n+(8/9)(10^n-1)+1
=(4/9)*10^2n-(4/9)*10^n+(8/9)*10^n-8/9+1
=(4/9)*10^2n+(4/9)*10^n+1/9
=(1/9)(4*10^2n+4*10^n+1)
=[(1/3)(10^n+1)]^2
是一个完全平方数
得证