Statement-1 The sum of the series 1 +(1+ 2+4)+ (4+ 6+ 9)+(9+12+16) ...(361 +380 +400) is 8000. Statement-2 `sum_(k=1)^n(k^3-(k-1)^3)=n^3`for any natural number n. (1) Statement-1 is true, Statement-2 is false. (2) Statement-1 is false, Statement-2 is true. (3) Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-l (4) Statement-1 is true, Statement-2 is true Statement-2 is not a correct explanation for Statement-l.

Explanation

`1 + (1 + 2 + 4) + (4 + 6 + 9) + (9+12+ 16) + ...`

`T_n = n^2 + n(n+1) + (n+1)^2`

`S= ((0)^2 + 0.1 + 1^2) + (1^2 + 1*2 + 2^2) + .... (19^2 + 19*20 + 20^2)`

`= sum_(n=0)^19 n^2 + (n)(n+1) + (n+1)^2 `

`= sum_(n=1)^20 ((n-1)^3 - n^3)/((n-1) - n)`

`= sum_(n=1)^20 n^3 - (n-1)^3 `

`S= (20^3 - 19^3) + (19^3 - 18^3) + (18^3 - 17^3) ..... (1^3 - 0^3) `

`= 20^3 - 0^3 = 8000`

...