An important shorthand notation for calculations.
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Concept 
Content 
Natural Numbers
and Induction 
Using natural numbers is our first mathematical abstraction as children. Mathematical induction is an important technique of proof. 
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We will use the following notations. \[\begin{aligned}
\sum_{i=1}^n a_i &= a_1+a_2+\dots+a_{n1}+a_n\\
\prod_{i=1}^n a_i &= a_1 \cdot a_2 \cdot \dots \cdot a_{n1}\cdot
a_n\\
\bigcup_{i=1}^n A_i &= A_1 \cup A_2 \cup \dots \cup A_{n1}\cup A_n
\end{aligned}\] The union works also for an arbitrary index set
\(\mathcal{I}\): \[\bigcup_{i \in \mathcal{I}} A_i = \{ x \,:\,
\exists i \in \mathcal{I} \text{ with } x \in A_i \} \,.\]
The first is a useful notation for a sum which is the result
of an addition. Two or more summands added. Instead of using
points, we use the Greek letter \(\sum\). For example, \[3+7+15+\ldots+127\] is not an unambiguous
way to describe the sum. Using the sum symbol, there is no confusion:
\[%\label{eq:sum1}
\sum_{i=2}^7 (2^i1).\] Of course, the parentheses are necessary
here. You can read this as a for
loop:
Concept 1 (for loop for the sum above). sum := 0;
for i:=2 to 7 do {
sum := sum + 2^i1);
}
Rule of Thumb 2. Let \(i\) run from \(2\) to \(7\), calculate \(2^i1\) and add.
index variable: 
\(i=2\), 
first summand: 
\(2^i1=2^21=\) 
\(41=\) 
\(3\) 
index variable: 
\(i=3\), 
second summand: 
\(2^i1=2^31=\) 
\(81=\) 
\(7\) 
index variable: 
\(i=4\), 
third summand: 
\(2^i1=2^41=\) 
\(161=\) 
\(15\) 
index variable: 
\(i=5\), 
fourth summand: 
\(2^i1=2^51=\) 
\(321=\) 
\(31\) 
index variable: 
\(i=6\), 
fifth summand: 
\(2^i1=2^61=\) 
\(641=\) 
\(63\) 
index variable: 
\(i=7\), 
last summand: 
\(2^i1=2^71=\) 
\(1281=\) 
\(127\) 


Sum: 


\(246\) 
Example 3. \[\sum\limits_{i=1}^{10} (2i1) = 1+3+5+\ldots+19
\stackrel?= 100\] \[\sum\limits_{i=10}^{10} i =
1098\ldots1+0+1+\dots+8+9+10 \stackrel?=0\]
With the same construction, we describe the result of a
multiplication, called a product, which consists of two or more
factors. There we use the Greek letter \(\prod\). For example: \[\prod_{i=1}^8 (2i) = (2\cdot 1)\cdot (2\cdot
2)\cdot (2\cdot 3)\cdot \ldots \cdot (2\cdot 8)
\stackrel?= 10321920.\]
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