Question

$\text{Statement 1}:\text{The sum of the series}1+(1+2+4)+(4+6+9)+....+(361+380+400)=8000$

$\text{Statement 2}$ :
$\sum _{k=1}^n(k^3-(k-1{)}^3)=n^3$, for any natural number, $n$

• A. statement $1$ is false, statement $2$ is true
• 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

Answer 1

The correct option is B statement $1$ is true , statement $2$ is true ; statement $2$ is a correct explanation for statement $1$

$\text{Statement 2}$ :
$\sum _{k=1}^n(k^3-(k-1{)}^3)=n^3$
L.H.S$=$
$1^3-0^3$
$+2^3-1^3$
$+\cdots \cdots$
$+n^3-(n-1{)}^3$
$=n^3$
So, Statement $2$ is correct

$\text{Statement 1}$ :
$1+(1+2+4)+(4+6+9)+........+(361+380+400)=8000$
$T_r=(r-1{)}^2+(r-1)r+r^2$
$=r^2+1-2r+r^2-r+r^2$
$=3r^2-3r+1$
$=r^3+3r^2-3r+1-r^3$
$=r^3-[r^3-3r^2+3r-1]$
$=r^3-(r-1{)}^3$
$\sum _{r=1}^{20}{T}_r=\sum _{r=1}^{20}[r^3-(r-1{)}^3]={20}^3=8000$

Hence statement $1$ is true , statement $2$ is true ; statement $2$ is a correct explanation for statement $1$