Prove the formula n(n+1) for the sum of first n even numbers.

The formula n(n+1) for the sum of the first nn even natural numbers can be proved using the concept of arithmetic progression (AP):

Step 1: Sequence Identification

The first nn even natural numbers are: 2,4,6,8,…,2n

• First term (a) = 2
• Common difference (d) = 2
• Number of terms (n)

Step 2: AP Sum Formula

The sum of the first nn terms of an AP is: S_n=n/2[2a+(n−1)d]

Step 3: Substitute Values

Substituting a=2, d=2

S_n=n/2[2×2+(n−1)×2]

=n/2[4+2n−2]

=n/2[2n+2]

=n/2×2(n+1)

=n(n+1)

Conclusion:

Thus, the sum of the first nn even natural numbers is:

S_n=n(n+1)

This confirms that n(n+1) is indeed the sum of the first n even natural numbers.