Appendix A: Mathematical Facts - Codes: The Guide to Secrecy from Ancient to Modern Times - page 471

we are adding the same number n times. The upper limit of summation tells us

how many times that is. Similarly, we can write, j =1 3=3+3+3+3=12.
This is the simplest application of the sigma notation. Another example is

10
i =1 i = 55.

Theorem A.5 ( Properties of the Summation (Sigma) Notation)
Let h,k,m,n

∈ Z with m ≤ n and h ≤ k .If R is a ring ( see page 483 ) ,

then:

R , then i = m ca i = c i = m a i .
(b) If a i ,b i ∈

(a) If a i ,c ∈

R , then i = m ( a i + b i )= i = m a i + i = m b i .

(c) If a i ,b j ∈ R , then

a i b j = n

n .

n k k k n k

=

a i b j a i b j = b j a i

i = m j = h i = m j = h j = h i = m j = h i = m

Proof . See page 599.

✷

Of value is the following formula.

Theorem A.6 (A Geometric Formula)
If a,r

∈ R

=1 , n

∈ N , then

r

n

ar j = a ( r n +1

− 1)

.

r − 1

j =0

Proof . See [169, Theorem A.30, page 283].

✷

A close cousin of the summation symbol is the following.

The Product Symbol
The multiplicative analogue of the summation notation is the product symbol
denoted by Π, upper case Greek pi . Given a m ,a m +1 ,...,a n ∈ R , where R is a

given ring and m ≤ n , their product is denoted by

n

···

a i = a m a m +1 a n ,

i = m

and by convention, i = m a i =1if m>n .
The letter i is the product index , m is the lower product limit , n is the upper
product limit , and a i is a multiplicand or factor .

For instance, i =1 i =1 · 2 · 3 · 4 · 5 · 6 · 7=5 , 040. This is an illustration of

the following concept.