Statement $1$: The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots \ldots+(361+380+ 400)$ is $8000$. Statement $2$: $\sum_{k=1}^n\left(k^3-(k-1)^3\right)=n^3$ for any natural number $n$.

• A

  Statement $1$ is false, statement $2$ is true.
• Correct answer
  B

  Statement $1$ is true, statement $2$ is true; statement $2$ is a correct explanation for statement $1$
• C

  Statement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation for statement $1$
• D

  Statement $1$ is true, statement $2$ is false

Hence, the correct answer is option: 'B'.