`S=1+(2.6.10.14...(4n-2))/((n+5).(n+6)....2n)`

`=1+(2^n [1.3.5.7...(2n-1)])/((n+5)(n+6)...2n)`

`=1+(2^n [1.2.3.4....(2n-1).(2n)])/(2.4.6.8...2n.(n+5)(n+6)...(2n))`

`=1+(1.2.3.4...(2n-1).(2n))/(1.2.3.4...n.(n+5).(n+6)...(2n))`

`=1+(n+1)(n+2)(n+3)(n+4)`

`=1+[(n+1)(n+4)][(n+2)(n+3)]`

`=1+(n^2 +5n+4)(n^2 +5n+6)`

`=1+(n^2 +5n+4)^2 +2(n^2 +4n+4)`

`=(n^2 +5n+5)^2` là số chính phương ( dpcm )