math
11th Grade - Sequences and induction

11th Grade lesson

5 - Sequences and induction


Vocabulary

You've seen in the 10th Grade lesson about the sequences what an increasing sequence is, a decreasing sequence, monotonic, bounded above, below, and both. Now, let's see what a convergent sequence is and what are adjacent sequences.

A convergent sequence approaches a certain number, called limit of the sequence, when n tends to infinity. It is a sequence u such that there is a real number l such that limites de suite.

Two adjacent sequences are such that one is increasing and the other decreasing and the terms of both are getting closer as n tends to infinity, that means sequence limit.

Examples :
- the sequence defined for every n by sequence is decreasing, monotonic, bounded above and below and convergent. Its limit is 2.
- the sequence defined for every n by sequence is bounded and divergent.

Notice that an increasing sequence is always bounded below by its first term, and a decreasing sequence is always bounded above by its first term. A monotonic sequence can be either convergent or divergent.

Properties

Every bounded above and increasing sequence is convergent and every decreasing and bounded below sequence is convergent (but be careful, their limit is not always equal to the majorant or minorant).
If two sequences are adjacent, then they are convergent and converge to the same limit.
Increasing majored sequence
Adjacent sequences
sequence
sequence

Sequences defined by induction

A sequence is defined by induction when you are given the value of one term and a relation between the general term of order n and the next term of order n+1. For example, the sequence sequence is defined by induction. f is the function that gives sequence in function of sequence. If you know that the sequence u is convergent and the function f is continuous in l , then, by passing to the limit in the induction relation, we obtain the equality equation. Generally, this equation enables us to calculate l.

Also, note that for two sequences defined this way, you can determine an approximate value of the terms of the sequence and make conjectures about the convergence of the sequence, thanks to a drawing. Then let's draw in an orthornormal frame the graph of f, and let's place on the X-axis the first term sequence. We have image de fonction then using the graph of f, let's place on the Y-axis the term sequence . Now let's draw the line of equation equation y=x. Come back, from sequence on this line, and go down on the X-axis, so that you put back sequence on the X-axis. Now, using the graph of f, place sequence on the Y-axis and put back its value on the X-axis thanks to the line y=x. This way you can place as many terms of the sequence on the X-axis as you want and guess the limit of the sequence.

sequence


Mathematical induction

Mathematical induction is a method of reasoning that enables us to prove that some property is true for every number n. For example, if you have to prove that recursion proof is always a multiple of 3, you will generally use induction. Induction consits in 4 steps.


1. Suppose math recursion="the property you want to prove", for example here you'll suppose recursion proof


2. Show that recursion proof is true. Generally it's rather simple. Here propriete is true because recursion proof and 0 is a multiple of 3.


3. Show that for every number n, if cours recursion is true, then recursion lesson is also true. It's the most difficult step. To write the solution, you will for example say : "n is a natural number. Suppose that recursion is true.". We want to show that recursion is also true, that means that recursion is a multiple of 3.
recursion

recursion induction example is a multiple of 3.
math is a multiple of 3 because math is true. The sum of two multiples of 3 is a multiple of 3, hence recursion induction example is a multiple of 3, thus recursion induction example is a multiple of 3, and therefore recursion induction example is true.


4. Then conclude. Since recursion conclusion is true, and since for every n, recursion conclusion, we have recursion conclusion, then recursion conclusion is true, recursion conclusion then recursion conclusion is true... and therefore recursion conclusion is true for every n. To say it, just write : "using the induction principle, recursion property is true for every n".



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Mathematical induction

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