If n is an odd integer greater than or equal to 1, the value of `=n^(3)-(n-1)^(3)+(n-2)^(3)-"..."+(-1)^(n-1)1^(3)` is

Options

A. `((n+1)^(2)(2n-1))/(4)`
B. `((n-1)^(2)(2n-1))/(4)`
C. `((n+1)^(2)(2n+1))/(4)`
D. none of these

Answer

A

Explanation

We have,
`n^(3)-(n-1)^(3)+(n-2)^(3)+ . . . . . . +(-1)^(n-1)1^(3)`
`=1^(3)-2^(3)+3^(3)-4^(3)+ . . .. +n^(3)" "[because" n is odd"]`
`=[1^(3)+2^(3)+3^(3)+4^(3)+ . . . +n^(3)]-2[2^(3)+4^(4)+ . . . .+(n-1)^(3)]`
`={(n(n+1))/(2)}^(2)-2^(4){1^(3)+2^(3)+ . . . . +((n-1)/(2))^(3)}`
`={(n(n+1))/(2)}^(2)-16{(1)/(2)((n-1)/(2))((n-1)/(2))((n-1)/(2)+1)}^(2)`
`={(n(n+1))/(2)}^(2)-(1)/(4){(n+1)(n-1)}^(2)`
`=((n+1)^(2))/(4){n^(2)-(n-1)^(2)}=((2n-1)(n+1)^(2))/(4)`