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प्रश्न
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then write the value of k.
उत्तर
According to the question,
\[\Rightarrow 2 \times \frac{n\left( n + 1 \right)}{2} = k\left[ \frac{n}{2}\left\{ 2 \times 1 + \left( n - 1 \right) \times 2 \right\} \right]\]
\[ \Rightarrow \frac{2n\left( n + 1 \right)}{2} = k\left[ \frac{n}{2}\left( 2 + 2n - 2 \right) \right]\]
\[ \Rightarrow n\left( n + 1 \right) = k\left[ \frac{n}{2}\left( 2n \right) \right]\]
\[ \Rightarrow n^2 + n = k n^2 \]
\[ \Rightarrow k = \frac{n + 1}{n}\]
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