Assertion (A) \[ (1+2+4)+(4+6+9)+(9+12+16)+\ldots+(81+90+100)=1000 \] Reason (R) \[ \sum_{r=1}^{n} \left(…
Assertion (A)
\[ (1+2+4)+(4+6+9)+(9+12+16)+\ldots+(81+90+100)=1000 \]
Reason (R)
\[ \sum_{r=1}^{n} \left( r^3-(r-1)^3 \right) = n^3 \]
for any natural number $n$
- Both \((A)\) and \((R)\) are true and \((R)\) is the correct explanation of \((A)\)
- Both (A) and (R) are true but (R) is not the correct explanation of (A)
- \((A)\) is true but \((R)\) is false
- (A) is false but (R) is true
Solution
Since,
$\begin{aligned}
& 1+(1+2+4)+(4+6+9)+(9+12+16) \\
& +\ldots+(81+90+100) \\
& =1+\left(1^2+(2 \times 1)+2^2\right)+\left(2^2+(2 \times 3)+3^2\right)+\left(3^2\right. \\
& \left.+(3 \times 4)+4^2\right)+\ldots+\left(9^2+(9 \times 10)+10^2\right) \\
& =\sum_{r=1}^{10}\left[(r-1)^2+r(r-1)+r^2\right] \\
& =\sum_{r=1}^{10}[r-(r-1)]\left[(r-1)^2+r(r-1)+r^2\right] \\
& =\sum_{r=1}^{10}\left[r^3-(r-1)^3\right] \\
& =\left(1^3-0^3\right)+\left(2^3-1^3\right)+\left(3^3-2^3\right)+\ldots+\left(10^3-9^3\right) \\
& =10^3-0^3=1000 \\
\end{aligned}$
So, both $A$ and $R$ are true and $R$ is the correct explanation of $A$. Hence, option $1$ is correct.
Asked in: AP EAMCET 2019 (20 Apr Shift 1)
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