Concrete mathematics : a foundation for computer science

30 SUMS
2.3 MANIPULATION OF SUMS Not to be confused
with finance.
The key to success with sums is an ability to change one t into
another that is simpler or closer to some goal. And it’s easy to do this by
learning a few basic rules of transformation and by practicing their use.
Let K be any finite set of integers. Sums over the elements of K can be
transformed by using three simple rules:
x cak = c pk;
(distributive law) (2.15)
kEK kEK
~iak+bk) = &+~bk; (associative law) (2.16)
kEK kEK UK
x ak = x %(k) * (commutative law) (2.17)
kEK p(k)EK
The distributive law allows us to move constants in and out of a t. The
associative law allows us to break a x into two parts, or to combine two x’s
into one. The commutative law says that we can reorder the terms in any way
we please; here p(k) is any permutation of the set of all integers. For example, Why not call it
if K = (-1 (0, +l} and if p(k) = -k, these three laws tell us respectively that permutative instead
of commutative?
ca-1 + cao + cal = c(a-j faofal); (distributive law)
(a-1 Sb-1) + (ao+b) + (al +bl)
= (a-l+ao+al)+(b-l+bo+bl); (associative law)
a-1 + a0 + al = al + a0 + a-1 . (commutative law)
Gauss’s trick in Chapter 1 can be viewed as an application of these three
basic laws. Suppose we want to compute the general sum of an arithmetic
progression,
S = x (afbk).
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