AP EAMCET · Maths · Sequences and Series
If \(4^3+8^3+12^3+\ldots\) up to \(n\) terms \(=k n^2(n+1)^2\) (for all \(n \in \mathrm{N}\)), then \(k=\)
- A 4
- B 8
- C 16
- D 32
Answer & Solution
Correct Answer
(C) 16
Step-by-step Solution
Detailed explanation
\(\sum_{i=1}^{n} (4i)^3 = 64 \sum_{i=1}^{n} i^3\) \(64 \left(\frac{n(n+1)}{2}\right)^2 = 64 \frac{n^2(n+1)^2}{4}\) \(16 n^2(n+1)^2 = k n^2(n+1)^2\) \(k = 16\)
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