Help on Math question

Question

What is the sum of all even natural numbers between 1 and some number N?

Answer 1

sum of all even natural numbers = 2+4+6 + ...+ 2(N-1) + 2N

= (N/2)(a1 +aN) where aN= a1 +d(N-1), d= common difference = 2, a1 = 2

SN = (N/2)(a1 + a1 +dN -d)

SN = (N/2)(2a1 + dN -d)

for example, sum of n terms with n = 6 is 2+4+6+8+10+12 = 3(2+12) = 3(14) = 42

= S6 = (6/2)(2(2)+ 2(6) -2) = 3(4+12-2) = 3(14) = 42