Question

To calculate tha sum of first n even natural numbers and verify the formula
Sn = n (n+1)
For the sum of first even natural numbers.

Answer 1

To find --> Sum of first n even natural numbers and verify the formula

Sₙ = n (n + 1 )

Solution--> Natural numbers are collection of whole number starting from one and even natural numbers are those natural number which are divisible by 2 .

So first n even natural numbers are

2, 4 , 6 , 8 ,...................... , 2 n

These number forms an AP whose

First term = 2

Common difference = a₂ - a₁

= 4 - 2 = 2

Now

a = 2 , d = 2 , n = n , Sₙ = ?

Sₙ = n / 2 { 2a + (n -1 ) d }

= n / 2 { 2 (2) + (n - 1 ) (2) }

= n/2 ( 4 + 2n - 2 )

= n / 2 ( 2 n + 2 )

=( n / 2) × 2 × ( n + 1 )

2 is cancel out from numerator and denominator and we get

Sₙ = n ( n + 1 )

This is same as formula given in question

So given formula is correct for sum of first even natural numbers