# Metric relations in any triangle : 11th grade math lesson

Numerical sequences in a 11th grade math lesson where we will approach the definition of a sequence and its direction of variation.
In this eleventh grade lesson, we will study two families of sequences, arithmetic and geometric sequences as well as their direction of variation according to the value of the reason. We will then finish with the calculation of the sum of the first n terms of a numerical sequence.

## I. Definition

Definition:

A numerical sequence is a function of in : .

Its definition set is therefore or a subset of .

## II. Notations – Vocabulary

Notations and vocabulary of sequences :

The variable n being a natural integer, this integer n allows to number the images: in addition to the classical functional writing s(n) used to designate the image of the natural integer n by the function s, we can also use the indexed notation:sn. With this notation the image of 0 is written: s0.

– With this notation, we say that :

1. s(n) =sn is the term of index n or rank n of the sequence s.
2. s is the sequence of general termsn and we write: s = (sn)
3. s(0) = s0 which is the image of 0 by s is also called term of rank 0 of the sequence s .
4. s(1) = s1 which is the image of 1 by s is also called rank 1 term of the sequence s.

If the numbering starts at rank 0, s(0) = s0 is the first term of the sequence s . s(1) = s1 is the second term of the sequence s.

It sometimes happens that the first term of a sequence s is not s0.

Example:

does not exist for n = 0. The sequence starts at rank 1. We will then write: for .

does not exist for n = 0, nor for n = 1. The suite starts at row 2.

In all cases of this type, we will specify the subset of where the sequence is defined: Here, we have: .

## III. Various ways to define a sequence

### 1. Suites defined by a function formula:

Definition:

For this, most of the time, we restrict to a function defined on or a subset of containing .

For example, the sequenceun = n2 ( ), is the restriction to n of the function f defined on by f(x) = x2.

Thus, the properties already studied for the functions of the real variable will be usable for the sequences!

However, we will also study some examples of sequences associated with functions that you have not yet studied in1st grade; for example, the geometric sequenceun = 2n is associated with the exponential function defined on by f(x) =2x, which will be studied in the last grade.

### 2. Suites defined by a recurrence formula:

For any natural number n, the image s(n) =sn is “numberable”.

Definition:

We can define the term of rank (n+1) according to the previous term of rank n by a formula called recurrence formula.

More precisely, the sequence s = (sn) will be defined by recurrence by

1. His first term .
2. – An equality connecting any two consecutive terms of the sequence:
3. where is a known function.

Example:

The sequence defined by its first term u0 = 4096 and the recurrence formula verified for any integer n: .

We obtain: u1 = = = 64

u2 = = = 8

u3 = =

u4 = =

u5 = …. and so on …

Here the function f is defined by .

## IV. Graphical representations of sequences

Definition:
When the sequence is defined by a functional equality of the type , the traditional representation of the graphs of function is usable: One obtains then the points of whole abscissas of the graph of the function of the real variable x: .

Example:

The graph of the suite defined on by: corresponds to the point of abscissa of the function defined on by .

When the sequence is defined by a recurrence formula of the type , this representation is not directly realizable.

A “spider’s web” type of representation is then used.

Example:

On the graph above, are drawn the lines of equation and y = x.

This device allows to visualize the successive terms of the sequence defined on by :

u0 = 10 and , for all :

Indeed:

## V. Monotonic sequences.

### 1. Direction of variation of a sequence.

 If for all $n\in\mathbb{N}$, we have: $U_n\leq\,\,U_{n+1}$ $U_n=\,U_{n+1}$ $U_n\,\geq\,\,U_{n+1}$ Direction of variation of $(U_n)$ $(U_n)$ growing $(U_n)$ constant $(U_n)$decreasing Absolute change $U_{n+1}-U_n\,\geq\,\,0$ $U_{n+1}-U_n\,=\,0$ $U_{n+1}-U_n\,\leq\,\,0$ Quotient (strictly positive terms) $\frac{U_{n+1}}{U_n\,}\,\geq\,\,1$ $\frac{U_{n+1}}{U_n\,}\,=\,1$ $\frac{U_{n+1}}{U_n\,}\,\leq\,\,1$

## VI. Bounded, major and minor sequences.

Same definitions as for functions of the real variable.

Example:

The sequencesn = sin n is a bounded sequence. Indeed: it is increased by 1 and decreased by (-1).

## VII. Periodic sequences.

Definition:

We say that a sequence is periodic of period , when, for any , we have: , p being the smallest non-zero natural number verifying this.

Example:

The constant sequences are periodic of period 1.

The sequence is periodic of period 2.

## VIII. Arithmetic sequences.

Definition:

When we go from any term of a sequence to the next term, always adding (or subtracting) the same number, we say that the sequence is arithmetic.

That is, if there exists , such that, for any , we have :

we say that the sequence is arithmetic of reason r.

The increases of an arithmetic sequence are thus constant: this constant is the reason r of the arithmetic sequence.

Examples:

1. The sequence of natural numbers is arithmetic with first term 0 and reason 1.
2. The sequence of even natural numbers is arithmetic with first term 0 and reason 2.
3. The sequence of odd natural numbers is arithmetic with first term 1 and reason 2.
4. The constant sequence of general termUn = 2 is arithmetic of first term 2 and reason 0.

## IX. Geometric sequences.

Definition:

When we go from any term of a sequence to the next term, always multiplying (or dividing) by the same non-zero number, we say that the sequence is geometric.

That is, if there exists , such that, for any , we have :

We then say that the sequence is geometric of reason .

The multiplier coefficients of a geometric sequence are therefore constant: this constant is the reason q of the geometric sequence.

The rates of increase of a geometric sequence are also constant. Indeed:

.

The geometric sequence of reason q has therefore a constant rate of increase t = q – 1.

Example:

The constant sequence of general termUn = 2 is geometric with first term 2 and reason 1.

The sequence of general termUn = (-1)n is geometric with first term U0 = 1 and reason -1.

Remark:

1. A sequence whose successive absolute variationsSn+1Sn = r are constant, i.e. independent of n, is an arithmetic sequence of reason r.
2. A sequence whose successive relative variations are constant, that is to say independent of n, is a geometric sequence of reason q = 1 + t.

For example, with a relative increase of t = 5% = 0.05 , then, q = 1.05 .

Indeed, if 0.05 , then :Sn+1Sn = 0.05Sn.

Therefore :Sn+1 =Sn + 0.05Sn = (1 + 0.05)Sn.

This gives:Sn+1 = 1.05Sn.

We have a geometric sequence of reason q = 1.05.

And in the general case, if , then: .

So: .

So we have a geometric sequence of reason .

## X. Arithmetic and geometric sequences: summary.

S is a sequence and n is any natural number:

 Arithmetic sequence of reason r Geometric sequence of reason q ¹ 0 recurrence formula $S_{n+1\,}=\,S_n\,+\,r$ $S_{n+1}\,=\,q\,S_n$ characterizations $S_{n+1\,}-\,S_n\,=\,r$ (constant) if $S_{0}\,\neq0$, $\frac{S-{n+1}}{S_n}=q$ (constant) term of rank n : function formula 1st term + n times the reason $S_n\,=\,S_0\,+\,n\,r$ $S_n\,=\,S_1\,+\,(n-1)\,r$ 1st term $\times$ reason exponent n $S_n\,=\,S_0\,\times \,q^n$ $S_n\,=\,S_1\times \,q^{n-1}$

## XI. Some interesting remarks.

### 1. Arithmetic sequences.

Definition:

The sequence defined by the formula: (affine function of ) is the arithmetic sequence of first term and reason . The graphical representation of an arithmetic sequence is thus formed of aligned points.

### 2. Geometric sequences.

Definition:

The sequence of powers of a non-zero real number a, of general term is

the geometric sequence with first term and reason .

The graphical representation of a geometric sequence of reason different from 1 is thus formed of points which are not aligned (they are located on an exponential curve).

### 4. Arithmetic sequences and direction of variation.

Ownership:

(un) is an arithmetic sequence of reason r.

1. If r > 0, then (un) is strictly increasing.
2. If r < 0, then (un) is strictly decreasing.
3. If r = 0, then (un) is constant.

### 5. Geometrical sequences and direction of variation.

Ownership:

(un) is a geometric sequence with reason and first term .

1. If q < 0, then (un) is not monotonic (the terms are alternately positive, then negative).
2. If q > 1 and if u0 > 0 , then (un) is strictly increasing.
3. If q > 1 and if u0 < 0 , then (un) is strictly decreasing.
4. If 0 < q < 1 and if u0 > 0 , then (un) is strictly decreasing.
5. If 0 < q < 1 and if u0 < 0 , then (un) is strictly increasing.
6. If q = 1, then (un) is constant.

### 6.arithmetic sequences and sum of consecutive terms.

Ownership:

If is an arithmetic sequence of reason r, then, for all , we have :

equality which is also written: .

To use this formula, it may be helpful to see that: .

In particular:

### 7. Geometric sequences and sum of consecutive terms.

Ownership:

If is a geometric sequence of reason , then, for all , we have :

equality which is also written :

.

To use this formula, it may be useful to see that :

.

In particular: .

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