AP EAMCET · Maths · Sequences and Series

Assertion (A)
\(\text { (I) }+(1+2+4)+(4+6+9) +(9+12+16)+\ldots+(81+90+100)=1000\)
Reason (R) \(\sum_{r=1}^n\left(r^3-[r-1]^3\right)=n^3\) for any natural number \(n\)

A Both \((A)\) and \((R)\) are true and \((R)\) is the correct explanation of \((A)\)
B Both (A) and (R) are true but (R) is not the correct explanation of (A)
C \((A)\) is true but \((R)\) is false
D (A) is false but (R) is true

Verified Solution

Answer & Solution

Correct Answer

(A) Both \((A)\) and \((R)\) are true and \((R)\) is the correct explanation of \((A)\)

Step-by-step Solution

Detailed explanation

Since,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Assertion (A)\(\text { (I) }+(1+2+4)+(4+6+9) +(9+12+16)+\ldots+(81+90+100)=1000\)Reason (R) \(\sum_{r=1}^n\left(r^3-[r-1]^3\right)=n^3\) for any natural number \(n\)

Step-by-step Solution

Assertion (A)
\(\text { (I) }+(1+2+4)+(4+6+9) +(9+12+16)+\ldots+(81+90+100)=1000\)
Reason (R) \(\sum_{r=1}^n\left(r^3-[r-1]^3\right)=n^3\) for any natural number \(n\)