Projects - Cengage Learning

C-112 Appendix C
EXAMPLE 1
Suppose x 1 , x 2 , . . . , x 25 are numbers such that a 30 and a 100.
25
k1
k1
Calculate a (x k 3) 2 .
k1
25
x k
25
2
x k
SOLUTION
25
25
a (x k 3) 2 a (x 2 k 6x k 9)
k1
k1
25
25 25
a x 2 k 6 a x k a 9
k1 k1 k1
100 6(30) 25(9) 505
Following, in summation notation, are three useful sums.
Useful Sums
n
n
n(n 1)
n(n 1)(2n 1)
1. a k 2. a k 2 3.
2
2
k1
k1
n
2
n(n 1)
a k 3 c d
2
k1
These three summation formulas can be proven using induction on the
number n of terms in the sum. An alternative proof of Useful Sum 1 is sketched
in Exercise 2.
Let’s put the properties of summation together with the useful sums to simplify
some sums.
EXAMPLE 2
50
Simplify a (3k 2).
k1
50
50 50
SOLUTION a (3k 2) a 3k a 2 Summation Property 2
k1
k1
50
3 a k 50(2)
k1
k1
50(50 1)
3c d 100
2
3725
Summation Properties 3 and 4
Useful Sum 1
EXAMPLE 3
25
Simplify a (2k 7) 2 .
k1