Question

 If the sum of first n even natural numbers is equal to K times the sum of first n odd natural numbers, then K is equal to?
A. $\dfrac{1}{n}$
B. $\dfrac{{n - 1}}{n}$
C. $\dfrac{{n + 1}}{{2n}}$
D. $\dfrac{{n + 1}}{n}$

Answer 1

Hint – We have to consider the n natural even and odd numbers and then we will find the sum of those n terms by using the formula and by putting the values in that formula. Then compare the sums of both A.P.s to find the value of K.

Complete step by step answer:
We know that,
First n even natural numbers are: 2,4,6,8.......
So we can see that they form an A.P. with first term = 2 and common difference = 2
Since, the formula of sum of n terms of an A.P. is
 ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ where,
a = first term
d = common difference
n = number of terms
Now, we substitute the values in this formula to get,
$
  Sum = \dfrac{n}{2}\left[ {2\left( 2 \right) + \left( {n - 1} \right)2} \right] \\
   = \dfrac{n}{2}\left[ {4 + 2n - 2} \right] = \dfrac{n}{2}\left[ {2 + 2n} \right] \\
 $
Now, we take 2 common and cancels it with the denominator
$ \Rightarrow Sum = n + {n^2}$
Since, we know that
First n odd natural numbers are: 1,3,5,7,9….
It also forms an A.P. with first term = 1 and common difference = 2
Again, we substitute these values in the formula of sum of n terms of an A.P.
$
  Sum = \dfrac{n}{2}\left[ {2\left( 1 \right) + \left( {n - 1} \right)2} \right] \\
   = \dfrac{n}{2}\left[ {2 + 2n - 2} \right] \\
 $
Simplify the above equation, we get
$ \Rightarrow Sum = \dfrac{{2{n^2}}}{2} = {n^2}$
According to the question,
$
  {n^2} + n = K \times {n^2} \\
   \Rightarrow {n^2}\left( {K - 1} \right) = n \\
 $
By further evaluating it
$
   \Rightarrow K - 1 = \dfrac{1}{n} \\
   \Rightarrow K = \dfrac{1}{n} + 1 = \dfrac{{n + 1}}{n} \\
 $
Therefore, the value of K is $\dfrac{{n + 1}}{n}$.
Hence, the correct option is D.

Note – It is to be noted that one should have the profound knowledge about the number system, so one can form the A.P.s used above and should have the capability to understand the language used in question to avoid mistakes. As students get confused, which of those two sums evaluated above have to multiply with K. And this is the only known approach to find the value of K.