College Algebra, 2013a

9.2 Summation Notation 665
S = n (2a +(n − 1)d)
2
If we rewrite the quantity 2a +(n − 1)d as a +(a +(n − 1)d) =a 1 + a n ,wegettheformula
( )
a1 + a n
S = n
2
A helpful way to remember this last formula is to recognize that we have expressed the sum as the
product of the number of terms n and the average of the first and n th terms.
To derive the formula for the geometric sum, we start with a geometric sequence a k = ar k−1 , k ≥ 1,
and let S once again denote the sum of the first n terms. Comparing S and rS, weget
S = a + ar + ar 2 + ... + ar n−2 + ar n−1
rS = ar + ar 2 + ... + ar n−2 + ar n−1 + ar n
Subtracting the second equation from the first forces all of the terms except a and ar n to cancel
out and we get S − rS = a − ar n .Factoring,wegetS(1 − r) =a (1 − r n ). Assuming r ≠1,wecan
divide both sides by the quantity (1 − r) to obtain
( ) 1 − r
n
S = a
1 − r
If we distribute a through the numerator, we get a − ar n = a 1 − a n+1 which yields the formula
S = a 1 − a n+1
1 − r
In the case when r =1,wegettheformula
S = a
}
+ a +
{{
...+ a
}
= na
n times
Our results are summarized below.