Projects - Cengage Learning

C-114 Appendix C
So
n3
n3 n3 n3
a (k 2) a k a 2 a k (n 2)2
k0
k0
k0
2
(n 3)(n 2)
2
(n 2)(n 1)
2
k1
(n 3)[(n 3) 1]
2(n 2)
2(n 2) n 2 [(n 3) 4]
2
n2 n 2
2
EXAMPLE 7
n
Simplify a k.
k4
SOLUTION
We’ll show two ways to simplify this sum.
The first method is to apply Useful Sum 1 directly.
The second method is to apply Useful Sum 1 by “shifting the index.”
Substitute i k 4. Then as k goes from 4 to n, i goes from 0 to n 4.
n n4
So a k a (i 4) substituting i k 4
k4
n4 n4
a i a 4
i0
i1
n n 3
a k a k a k (Right?)
k4
i0
k1
2
(n 3)(n 4)
2
n(n 1)
2
k1
3(3 1)
2
as in Example 6
(n 4)[(n 4) 1]
(n 3)4
n2 n 12
2
n2 n 12
2
(n 3)
[(n 4) 8]
2
Exercises
n
n
n
1. Derive Summation Property 2: a (a k b k ) a a k a b k .
k1
k1 k1
n
n(n 1)
2. Derive Useful Sum 1, a k , by writing the summation twice,
k1 2
the second time with the terms in reversed order, as in the derivation of the
formula for the sum of an arithmetic series, then adding.
15
15
15
2
3. Given, a x 30 and a x 50, calculate a (2x k 5) 2 k
k .
k1
k1
k1