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 $1$: 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 $n$.
Statement $1$ is false, statement $2$ is true.
Statement $1$ is true, statement $2$ is true; statement $2$ is a correct explanation for statement $1$
Statement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation for statement $1$
Statement $1$ is true, statement $2$ is false
Solution
Statement $1$ has $20$ terms whose sum is $8000$
And statement $2$ is true and supporting statement $1$.
$\because k^{\text {th }}$ bracket is $(k-1)^2+k(k-1)+k^2=3 k^2-3 k+1$.