If n is a positive integer satisfying the equation `2+(6*2^(2)-4*2)+(6*3^(2)-4*3)+"......."+(6*n^(2)-4*n)=140` then the value of n is

Explanation

`:.2+(6*2^(2)-4*2)+(6*3^(2)-4*3)+"......."+(6*n^(2)-4*n)=140`
`2+6(2^(2)+3^(2)+"......."+n^(2))-4*(2+3+"....."+n)=140`
`implies 2+6((n(n+1)(2n-1))/(6)-1)-4((n(n+1))/(2)-1)=140`
`implies 2+n(n+1)(2n+1)-6-2n(n+1)+4=140`
`implies n(n+1)(2n+1)-2n(n+1)-140=0`
`implies 2n^(3)3n^(2)+n-2n^(2)-2n-140=0`
`implies 2n^(3)+n^(2)-n-140=0`
`implies (n-4)+(2n^(2)+9n+35)=0`
`implies n=4 " or " 2n^(2)+9n+35=0`
`implies 2n^(2)+9n+35=0`
`implies n=(-9pmsqrt(81-280))/(4)`
`:. n=(9pmsqrt(-199))/(4) " " [" complex values "]`
Only positive integer value of n is 4.