Question

If the sum of first n even natural numbers is 420, find the value of n.

Answer 1

Hint: Here we go through by writing the series of first n even natural numbers as 2,4,6,8,…….n and we saw that this is the series of A.P. so apply the formula of summation of n terms of A.P. to solve our question.

Complete step-by-step answer:
Here in the question it is given that sum of n natural numbers is 420 that means,
2+4+6+8+…….up to n terms=420.
Here we clearly see that the series we write is in the form of A.P. with first term 2 and with a common difference of 2. So we apply the formula of summation of n terms of A.P.
Here first term (a) =2 and common difference (d) =2 and total sum ${S_n} = 420$
And we know that summation of n terms of A.P, ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Here put the given value in the formula to find out the value of n.
$
   \Rightarrow 420 = \dfrac{n}{2}\left[ {2 \times 2 + \left( {n - 1} \right) \times 2} \right] \\
   \Rightarrow 420 = n\left( {2 + n - 1} \right) \\
   \Rightarrow n(n + 1) = 420 \\
   \Rightarrow {n^2} + n - 420 = 0 \\$
To find the value of n we will solve the above quadratic equation using splitting middle term.
  $
   \Rightarrow {n^2} + 21n - 20n - 420 = 0 \\
   \Rightarrow n(n + 21) - 20(n + 21) = 0 \\
   \Rightarrow n = 20, - 21 \\
 $
Here n cannot be negative.
Therefore the value of n=20.

Note: Whenever we face such a type of question the key concept for solving the question is first always write the series with some steps and find out from which series it belongs then apply the formula of that series to find out the answer of the question. We should have knowledge about natural numbers as well.