U2. Sequences

Temporalización estimada: 7 horas. Examen, junto a la unidad 1, el 18 de octubre.

Temporalización: 7 horas

1. What is a sequence?
2. Some interesting sequences
3. Arithmetic progressions and geometric progressions
4. Limit of a sequence
5. Some important limits

Criterios de evaluación y estándares de aprendizaje evaluables.

Problema introductorio

Se desea organizar un torneo de ajedrez en los recreos. Al mismo se han apuntado 24 personas. Se jugará por el sistema de liguilla. ¿Dará tiempo a terminarlo antes de las vacaciones de Navidad?

Some famous and curious sequences

Prime numbers

2, 3, 5, 7, 11, 13, ...

Fibonacci

1, 1, 2, 3, 5, 8, 13, 21, ...

Un par de sucesiones extrañas

2, 10, 12, 16, 17, 18, 19, 200, ...

3, 3, 4, 6, 5, 4, 5, ...

Toothpic sequence

1, 3, 7, 11, 15, 27, 35, ...

Self-described sequence

1, 11, 21, 1211, 111221, 312211, ...

Search for them in the OEIS. The Online Enciclopedia of Integer Sequences.

Definition of sEquence

A sequence is an enumerated collection of numbers in which order matters. The collection is infinite.

a_1, a_2, a_3, ..., a_n, ...

You can define a sequence by:

Giving a explicit rule to get new terms. For example: "the ordered set of cubed natural numbers"
Giving a list of some terms of the sequence: 1, 8, 27, 64, 125, ....
An algebraic expression involving n: a_n = n³
Recurring law: a_n = a_(n-1) + a_(n-2)

Activities

Get the first five terms of the following sequences:

Arithmetic sequences

An arithmetic sequence is that where every term is got by adding the same number, called the difference, to the term before.

For example, the sequence of odd numbers: 1, 3, 5, 7, 9, ... can be seen as an arithmetic sequence because it starts at a_1=1 and the successive terms are appearing by adding d=2 to the one before.

Geometric sequences

A geometric sequence is that where every term is got by multiplying the same number, called the ratio, to the term before.

For example, the sequence of the powers of two, 1, 2, 4, 8, 16, 32, ... can be seen as a geometric sequence because it starts at 1 and the successive terms are appearing by multipliying by r=2 the one before.

The general term of arithmetic an geometric sequences can be seen in the attached image

2. Identify the following sequences as arithmetic or geometric; find their difference (d) or ratio (r) and their general term:

a. 3, 7, 11, 15, 19...; b. 3, 4, 6, 9, 13, 18, …; c) 3, 6, 12, 24, 48, …; d) 1, 3, 9, 27, 81, …; e) 5, –5, 5, –5, 5, …; f) 10, 7, 4, 1, –2, …; g) 100, 50, 25, 12.5, … h) 12, 12, 12, 12, …; i) 3, –5, 7, –9, 11, … j) 2 840; 284; 28.4; …; k) 90, –30, 10, –10/3, 10/9, … l) 17,4; 15,8; 14,2; 12,6; …

Sum of the terms of a sequence

Find the attached image for the formulae

Aplicaciones

Interés compuesto.
The terms of the following sequence tend to zero when n goes higher and higher. That is called “the limit” of the sequence. Give the absolute error between the 1000th term and that limit (b_n = -3/(2n+1))

Review about sequences (worksheet)

Download the following worksheet about sequences.

Word problems

Your teacher offers you a deal. Every class-day he will give you 1000 €. You will have to give him, in exchange, 1 cent. the first class-day, 2 cents. the second class-day and keep on doubling the quantity till we get to the end of this term. Would you accept the deal?
A worker earns a salary of 1200 €. In her contract is stipulated that the salary will increase every three years by 150 €. If she works for the company 35 years. Can you tell what will be her salary at the end of her lifework? What is the total amount of money earn by that worker?
You save 5000 € in a bank that offers you 4% of interest per year. How much will you have saved after ten years? How long will it take to get to 10000 €?

To get more of it

What is the 1000th term of the following sequence? 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5... (source).

Explanation.

Exam questions (Units 1 & 2)

Choose 9 out of the following 10 questions

1. About the set of Real Numbers:

- Classifying numbers written in different forms (fractions, decimals, radicals, scientific notation, factorials...).
- Sorting different kind of numbers. You will have to say to which set of numbers they belong: N, Z, Q, R...

2. Rationalising and/or simplifying expressions.

3. Find x in simple logarithmic equations.

4. Write something equals to n (n ∈ Z).

5. Factorial and combinatorics. Simplifying expressions.

6. Binomial expansion (find a specific term or expand a binomial).

7. Arithmetic or geometric sequences:

- General term.
- Sum of some terms.
- Sum of all the terms (if possible).

8. Errors (absolute, relative).

9. Easy problem.

10. Funny, new, different problem (FNDP)

Saberes básicos implicados en la unidad

A. Sentido numérico.

MATE.1.A.1. Sentido de las operaciones.

MATE.1.A.1.2 Estrategias para operar (suma, producto, cociente, potencia, radicación y logaritmo) con números reales y complejos: cálculo mental o escrito en los casos sencillos y con herramientas tecnológicas en los casos más complicados.

D. Sentido algebraico.

MATE.1.D.1. Patrones. Generalización de patrones en situaciones sencillas.

MATE.1.D.1.1. Generalización de patrones en situaciones sencillas.

MATE.1.D.5. Pensamiento computacional.

MATE.1.D.5.1 Formulación, resolución y análisis de problemas de la vida cotidiana y de la ciencia y la tecnología empleando herramientas o programas más adecuados.

MATE.1.D.5.2 Comparación de algoritmos alternativos para el mismo problema mediante el razonamiento lógico.

F. Sentido socioafectivo.

MATE.1.F.1. Creencias, actitudes y emociones.

MATE.1.F.1.2 Tratamiento del error, individual y colectivo, en el aula de matemáticas como elemento movilizador de saberes previos adquiridos y generador de oportunidades de aprendizaje en el aula de matemáticas.

MATE.1.F.2. Trabajo en equipo y toma de decisiones.

MATE.1.F.2.1 Reconocimiento y aceptación de diversos planteamientos en la resolución de problemas y tareas efectivas para el éxito en el aprendizaje de las matemáticas, transformando los enfoques de las y los demás en nuevas y mejoradas estrategias propias, mostrando empatía y respeto en el proceso.

Criterios de evaluación por unidades, competencias clave y estándares de aprendizaje evaluables

Criterios de evaluación

5. Calcular el término general de una sucesión, monotonía y cota de la misma. CMCT.

Estándares de aprendizaje evaluables

1.3. Utiliza la notación numérica más adecuada a cada contexto y justifica su idoneidad.

1.4. Obtiene cotas de error y estimaciones en los cálculos aproximados que realiza valorando y justificando la necesidad de estrategias adecuadas para minimizarlas.

Refuerzo

Quizizz.

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