sumofarithmeticseries

Sum of Arithmetic Series

Suppose S=SUM(1 to n) a+bn

S=n(a + b(n+1)/2)

Proof:

Basis for induction:

n=1 => S=a+b

Inductive hypothesis:

SUM(1 to n+1) a+bn=SUM(1 to n) a+bn +a+b(n+1)

=n(a + b(n+1)/2) + a+b(n+1)

=(n+1)(a + b(n/2 + 1))

=(n+1)(a + b((n+2)/2))