Difference between revisions of "An Introduction to Mathematical Induction: The Sum of the First n Natural Numbers, Squares and Cubes."

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(Created page with "== Sigma Notation == In math, we frequently deal with large sums. For example, we can write :: <math style="vertical-align: 0px">1+2+3+4+5+6+7+8+9+10+11+12+13, </math> whic...")
 
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In math, we frequently deal with large sums. For example, we can write
 
In math, we frequently deal with large sums. For example, we can write
 +
<br>
  
 
:: <math style="vertical-align: 0px">1+2+3+4+5+6+7+8+9+10+11+12+13, </math>
 
:: <math style="vertical-align: 0px">1+2+3+4+5+6+7+8+9+10+11+12+13, </math>
Line 10: Line 11:
 
:: <math style="vertical-align: 0px">1+2+\cdots+13. </math>
 
:: <math style="vertical-align: 0px">1+2+\cdots+13. </math>
  
However, there is an even more powerful shorthand known as '''sigma
+
However, there is an even more powerful shorthand known as '''sigma notation'''. When we write  
notation'''. When we write  
 
  
 
:: <math style="vertical-align: 0px">{\displaystyle \sum_{i=1}^{13}\, i,} </math>  
 
:: <math style="vertical-align: 0px">{\displaystyle \sum_{i=1}^{13}\, i,} </math>  
  
 
this means the same thing as the previous two mathematical statements.
 
this means the same thing as the previous two mathematical statements.
Here, the '''index''' below the capital sigma, <math style="vertical-align: 0px">\Sigma </math>, is the letter
+
Here, the '''index''' below the capital sigma, <math style="vertical-align: -5px">\left(\Sigma\right) </math>, is the letter <math style="vertical-align: 0px">i </math>, and the <math style="vertical-align: 0px">i </math> that follows the <math style="vertical-align: 0px">\Sigma </math> is our '''rule''' to
</math>i </math>, and the <math style="vertical-align: 0px">i </math> that follows the <math style="vertical-align: 0px">\Sigma </math> is our '''rule''' to
+
apply to each value of <math style="vertical-align: 0px">i </math>. The values <math style="vertical-align: -1px">1 </math> and <math style="vertical-align: -1px">13 </math> tell us how many times to repeat the rule, i.e., to follow the rule for <math style="vertical-align: -4px">i=1, </math>
apply to each value for <math style="vertical-align: 0px">i </math>. The values <math style="vertical-align: 0px">1 </math> and <math style="vertical-align: 0px">13 </math> tell us how
+
then add the rule for <math style="vertical-align: -4px">i=2, </math> then for <math style="vertical-align: -4px">i=3, </math> and continue in this
many times to repeat the rule, i.e., to follow the rule for <math style="vertical-align: 0px">i=1, </math>
+
manner until you reach <math style="vertical-align: -1px">13</math>. In other words,  
then add the rule for <math style="vertical-align: 0px">i=2, </math> then for <math style="vertical-align: 0px">i=3, </math> and continue in this
 
manner until you reach 13. In other words,  
 
  
 
:: <math style="vertical-align: 0px">{\displaystyle \sum_{i=1}^{13}}\, i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13. </math>  
 
:: <math style="vertical-align: 0px">{\displaystyle \sum_{i=1}^{13}}\, i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13. </math>  

Revision as of 22:28, 21 August 2015

Sigma Notation

In math, we frequently deal with large sums. For example, we can write

which is a bit tedious. Alternatively, we may use ellipses to write this as

However, there is an even more powerful shorthand known as sigma notation. When we write

this means the same thing as the previous two mathematical statements. Here, the index below the capital sigma, , is the letter , and the that follows the is our rule to apply to each value of . The values and tell us how many times to repeat the rule, i.e., to follow the rule for then add the rule for then for and continue in this manner until you reach . In other words,

Of course, we can change the rule and/or the index. For example,

Most importantly, we frequently don't have the luxury of bounds that are actual values. We can also write something like

or

These non-fixed indices allow us to find rules for evaluating some important sums.

Proof by (Weak) Induction

When we count with natural or counting numbers (frequently denoted

</math>\mathbb{N} </math>), we begin with one, then keep adding one unit at a

time to get the next natural number. In other words,

``The Natural Numbers </math>\,=\,\mathbb{N}\,=\,\{1,2,3,\ldots\}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots\}. </math>

This is the basis for weak, or simple induction; we must first prove our conjecture is true for the lowest value (such as 1), and then show whenever it's true for an arbitrary it's true for the as well. This mimics our expression of the natural numbers as ``just keep adding one.

It is also equivalent to prove that whenever the conjecture is true for it's true for Which approach you choose can depend on which is more convenient, or frequently which is more appealing to the teacher grading the work.

Although we won't show examples here, there are induction proofs that require strong induction. This occurs when proving it for the

</math>n^{\mathrm{th}} </math> case requires assuming more than just the 

case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value than down to our base case.

The Sum of the first Natural Numbers

Claim. The sum of the first natural numbers is

Proof. We must follow the guidelines shown for induction arguments. Our base step is and plugging in we find that

This gives us our starting point. For the induction step, let's assume the claim is true for so

Now, we have

as required.


The Sum of the first Squares

Claim. The sum of the first squares is

Proof. Again, our base step is and plugging in we find that

This gives us our starting point. For the induction step, let's assume the claim is true for so

Now, we have

as required.


The Sum of the first Cubes

Claim. The sum of the first cubes is

Notice that the formula is really similar to that for the first natural numbers.

Proof.. Plugging in we find that

completing our base step.

For the induction step, let's assume the claim is true for so

Now, we have

as required.


Aside from being good examples of simple or weak induction, these formulas are frequently used to find an integral as a limit of a Riemann sum. \end{document}