Sequels : 11th grade math lesson to download in PDF.

11th grade math lessons

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 (U_n) is a function of $\mathbb{N}$ in $\mathbb{R}$ : $n\,\mapsto \,U_n$ .

Its definition set is therefore $\mathbb{N}$ or a subset of $\mathbb{N}$ .

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 :

s(n) =_sn is the term of index n or rank n of the sequence s.
s is the sequence of general term_sn and we write: s = (_sn)
s(0) = _s0 which is the image of 0 by s is also called term of rank 0 of the sequence s .
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:

$S_n=\frac{1}{n}$ does not exist for n = 0. The sequence starts at rank 1. We will then write: for $n\in\,\mathbb{N}^*$ .

$t_n=\frac{1}{n(n-1)}$ 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 $\mathbb{N}$ where the sequence is defined: Here, we have: $n\in\mathbb{N}\setminus\,\,\{\,0;1\,\,\}$ .

III. Various ways to define a sequence

1. Suites defined by a function formula:

Definition:

For this, most of the time, we restrict to $\mathbb{N}$ a function defined on $\mathbb{R}$ or a subset of $\mathbb{R}$ containing $\mathbb{N}$ .

For example, the sequence_un = ⁿ² ( $n\in\mathbb{N}$ ), is the restriction to n of the function f defined on $\mathbb{R}$ 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 in^{1st grade}; for example, the geometric sequence_un = ²ⁿ is associated with the exponential function defined on $\mathbb{R}$ 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 $s(n+1)\,=\,s_{n+1}$ of rank (n+1) according to the previous term $s(n)\,=\,s_n$ of rank n by a formula called recurrence formula.

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

His first term $s(0)\,=\,s_0$ .
– An equality connecting any two consecutive terms of the sequence:
$s_{n+1\,}\,=\,f(s_n\,)$ where is a known function.

Example:

The sequence defined by its first term _u0 = 4096 and the recurrence formula verified for any integer n: $U_{n+1}=\sqrt{U_n}$ .

We obtain: _u1 = $\sqrt{U_0}$ = $\sqrt{4096}$ = 64

_u2 = $\sqrt{U_1}$ = $\sqrt{64}$ = 8

_u3 = $\sqrt{U_2}$ = $\sqrt{8}$

_u4 = $\sqrt{U_3}$ = $\sqrt{\,\sqrt{8}}$ $\simeq\,1,68$

_u5 = $\sqrt{U_4}$ $\simeq\,1,3$ …. and so on …

Here the function f is defined by $f(x)=\sqrt{x}$ .

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: $x\,\mapsto \,f(x)$ .

Example:

The graph of the suite defined on $\mathbb{N}^*$ by: $S_n=\frac{1}{n}$ corresponds to the point of abscissa $x\in\mathbb{N}^*$ of the function defined on $\mathbb{R}^*$ by $f(x)=\frac{1}{x}$ .

When the sequence is defined by a recurrence formula of the type $U_{n+1}=f(U_n)$ , 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 $y=\frac{1}{2}x+1$ and y = x.

This device allows to visualize the successive terms of the sequence defined on $\mathbb{N}$ by :

_u0 = 10 and , for all $n\in\mathbb{N}$ : $U_{n+1}=\frac{1}{2}U_n+1$

Indeed: $U_{1}=\frac{1}{2}U_0+1=\frac{1}{2}\,\times \,10+1=5+1=6$

$U_{2}=\frac{1}{2}U_1+1=\frac{1}{2}\,\times \,6+1=3+1=4$

$U_{3}=\frac{1}{2}U_2+1=\frac{1}{2}\,\times \,4+1=2+1=3$

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	growing	constant	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 sequence_sn = 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 (U_n) is periodic of period $p\in\mathbb{N}$ , when, for any $n\in\mathbb{N}$ , we have: $U_{n+p}=U_n$ , p being the smallest non-zero natural number verifying this.

Example:

The constant sequences are periodic of period 1.

The sequence $U_{n}=(-1)^n$ 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 $r\in\mathbb{Z}$ , such that, for any $n\in\mathbb{N}$ , we have :

$S_{n+1}=S_n+r$ we say that the sequence $S_{n}$ 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:

The sequence of natural numbers is arithmetic with first term 0 and reason 1.
The sequence of even natural numbers is arithmetic with first term 0 and reason 2.
The sequence of odd natural numbers is arithmetic with first term 1 and reason 2.
The constant sequence of general term_Un = 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 $q\in\mathbb{R}^*$ , such that, for any $n\in\mathbb{N}$ , we have :

$S_{n+1}=q\times \,S_n$ We then say that the sequence $(S_{n}\,)$ is geometric of reason $q\neq0$ .

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:

$\frac{S_{n+1}-S_n}{S_n}=\frac{S_n\times \,q-S_n}{S_n}=\frac{(q-1)S_n}{S_n}=q-1$ .

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

Example:

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

The sequence of general term_Un = (-1⁾ⁿ is geometric with first term _U0 = 1 and reason -1.

Remark:

A sequence $(S_{n}\,)$ whose successive absolute variations_Sn+1 –_Sn = r are constant, i.e. independent of n, is an arithmetic sequence of reason r.
A sequence $(S_{n}\,)$ whose successive relative variations $\frac{S_{n+1}-S_n}{S_n}=t$ are constant, i.e. 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 $\frac{S_{n+1}-S_n}{S_n}=$ 0.05 , then :_Sn+1 –_Sn = 0.05_Sn.

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

This gives:_Sn+1 = 1.05_Sn.

We have a geometric sequence of reason q = 1.05.

And in the general case, if $\frac{S_{n+1}-S_n}{S_n}=t$ , then: $S_{n+1}\,-\,S_n\,=\,t\,S_n$ .

So: $S_{n+1\,}=\,S_n\,+\,t\,S_n\,=\,(1\,+\,t)\,S_n$ .

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 U_0=b 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 $U_n\,=\,a^n$ is

the geometric sequence with first term $U_0\,=\,1$ 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).

3. Graphic illustrations.

4. Arithmetic sequences and direction of variation.

Ownership:

(_un) is an arithmetic sequence of reason r.

If r > 0, then (_un) is strictly increasing.
If r < 0, then (_un) is strictly decreasing.
If r = 0, then (_un) is constant.

5. Geometrical sequences and direction of variation.

Ownership:

(_un) is a geometric sequence with reason $q\,\neq\,0$ and first term $u_0\,\neq\,0$ .

If q < 0, then (_un) is not monotonic (the terms are alternately positive, then negative).
If q > 1 and if _u0 > 0 , then (_un) is strictly increasing.
If q > 1 and if _u0 < 0 , then (_un) is strictly decreasing.
If 0 < q < 1 and if _u0 > 0 , then (_un) is strictly decreasing.
If 0 < q < 1 and if _u0 < 0 , then (_un) is strictly increasing.
If q = 1, then (_un) is constant.

6.arithmetic sequences and sum of consecutive terms.

Ownership:

If (U_n) is an arithmetic sequence of reason r, then, for all $n\in\mathbb{N}$ , we have :

$U_0+U_1+U_2+....+U_n=(n+1)\frac{U_0+U_n}{2}$ equality which is also written: $\sum_{k=0}^{k=n}U_k=(n+1)\,\frac{U_0+U_n}{2}$ .

To use this formula, it may be helpful to see that: $(n+1)\,\frac{U_0+U_n}{2}=(n+1)\,(U_0+\frac{nr}{2}\,\,)$ .

In particular:

$1+2+3+4+....+n=\frac{n(n+1)}{2}$

7. Geometric sequences and sum of consecutive terms.

Ownership:

If (U_n) is a geometric sequence of reason $q\neq\,1$ , then, for any $n\in\mathbb{N}$ , we have :

$U_0+U_1+U_2+...+U_n=\frac{U_{n+1}-U_0}{q-1}$ equality which is also written :

$\sum_{k=0}^{k=n}U_k=\frac{U_{n+1}-U_0}{q-1}$ .

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

$\frac{U_{n+1}-U_0}{q-1}=U_0\,(\,\frac{q^{n+1}-1}{q-1}\,\,)$ .

In particular: $1+q+q^2+q^3+....+q^n=\frac{q^{n+1}-1}{q-1}$ .

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Numerical sequences : 11th grade math lesson in PDF

I. Definition

II. Notations – Vocabulary

III. Various ways to define a sequence

1. Suites defined by a function formula:

2. Suites defined by a recurrence formula:

IV. Graphical representations of sequences

V. Monotonic sequences.

1. Direction of variation of a sequence.

VI. Bounded, major and minor sequences.

VII. Periodic sequences.

VIII. Arithmetic sequences.

IX. Geometric sequences.

X. Arithmetic and geometric sequences: summary.

XI. Some interesting remarks.

1. Arithmetic sequences.

2. Geometric sequences.

3. Graphic illustrations.

4. Arithmetic sequences and direction of variation.

5. Geometrical sequences and direction of variation.

6.arithmetic sequences and sum of consecutive terms.

7. Geometric sequences and sum of consecutive terms.

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