If the sum of first n even natural numbers is equal to k times

In the given problem, we are given that the sum of the first n even natural numbers is equal to k times the sum of first n odd natural numbers.

We need to find the value of k

Now, we know that the first odd natural number is 1. Also, all the odd terms will form an A.P. with the common difference of 2.

So here,

First term (a) = 1

Common difference (d) = 2

So, let us take the number of terms as n

Now, as we know,

$S_{n}=\frac{n}{2}[2 a+(n-1) d]$

So, for n terms,

$S_{n}=\frac{n}{2}[2(1)+(n-1) 2]$

$=\frac{n}{2}[2+2 n-2]$

$=\frac{n}{2}(2 n)$

$=n^{2}$.........(1)

Also, we know that the first even natural number is 2. Also, all the odd terms will form an A.P. with the common difference of 2.

So here,

First term (a) = 2

Common difference (d) = 2

So, let us take the number of terms as n

So, for n terms,

$S_{n}=\frac{n}{2}[2(2)+(n-1) 2]$

$=\frac{n}{2}[4+2 n-2]$

$=\frac{n}{2}(2+2 n)$

Solving further, we get

$=n(1+n)$

$=n^{2}+n$..........(2)

Now, as the sum of the first n even natural numbers is equal to k times the sum of first n odd natural numbers

Using (1) and (2), we get

$n^{2}+n=k n^{2}$

$k=\frac{n^{2}+n}{n^{2}}$

$k=\frac{n(1+n)}{n^{2}}$

$k=\frac{n+1}{n}$

Therefore, $k=\frac{n+1}{n}$

Hence, the correct option is (d).