College Algebra & Trigonometry, 2018a

6.4. SUM OF A SERIES 299
Formulas for the sum of arithmetic and geometric series:
Arithmetic Series: like an arithmetic sequence, an arithmetic series has a constant
difference d. If we write out the terms of the series:
n∑
a k = a 1 + a 2 + a 3 + ···+ a n
k=1
we can rewrite this in terms of the first term (a 1 ) and the constant difference d.
n∑
a k = a 1 +(a 1 + d)+(a 2 +2d)+···+(a 1 +(n − 1)d)
k=1
This expression is equivalent to:
n∑
a k =(a 1 + a 1 + a 1 + ···+ a 1 )+(d +2d +3d + ···(n − 1)d)
k=1
n∑
a k = na 1 + d(1+2+3+···(n − 1))
k=1
Using the previous formula for the sum 1+2+3+···+(n − 1) gives us:
n∑
( ) (n − 1)n
a k = na 1 + d
2
k=1
This formula is often stated in various forms:
n∑
a k = n 2 (2a 1 +(n − 1)d)
k=1
or
n∑
a k = n 2 (a 1 + a n )
k=1
since a 1 +(n − 1)d = a n .