Question

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.

Answer 1

According to the question,

\[2 + 4 + . . . + 2n = k\left( 1 + 3 + 5 + 7 + . . . + \left( 2n - 1 \right) \right)\]

\[\Rightarrow 2 \times \frac{n\left( n + 1 \right)}{2} = k\left[ \frac{n}{2}\left\{ 2 \times 1 + \left( n - 1 \right) \times 2 \right\} \right]\]

\[ \Rightarrow \frac{2n\left( n + 1 \right)}{2} = k\left[ \frac{n}{2}\left( 2 + 2n - 2 \right) \right]\]

\[ \Rightarrow n\left( n + 1 \right) = k\left[ \frac{n}{2}\left( 2n \right) \right]\]

\[ \Rightarrow n^2 + n = k n^2 \]

\[ \Rightarrow k = \frac{n + 1}{n}\]