AP EAMCET · Maths · Mathematical Induction
\(\sum_{k=1}^{2 n+1}(-1)^{k-1} \cdot k^2\) equals to
- A \((n-1)(2 n-1)\)
- B \((n+1)(2 n+1)\)
- C \((n+1)(2 n-1)\)
- D \((n-1)(2 n+1)\)
Answer & Solution
Correct Answer
(B) \((n+1)(2 n+1)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \sum_{k=1}^{2 n+1}(-1)^{n-1} k^2 \\ & =\left[1^2-2^2+3^2-4^2+\ldots-(2 n)^2+(2 n+1)^2\right] \\ & =\left[1^2+3^2+5^2+\ldots+(2 n+1)^2\right] \\ & -\left[2^2+4^2+6^2+\ldots+(2 n)^2\right] \\ & =\left[1^2+2^2+3^2+4^2+\ldots+(2 n+1)^2\right] \\ &…
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