U7. Sequences

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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, 23, 35, 43, 55, 79, 95...

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.

How will the exam be like?

It will be a test type exam with 16 questions and five possible answers per question. In that test you will have to prove that you know how to do the following:

Identify some important sequences of numbers: even and odd numbers, prime numbers, Fibonacci, squares, cubes, triangular numbers, arithmetic, geometric...
Calculate the next number in one sequence.
Find any term of a sequence by knowing it's general term and perform a substitution.
Find the place knowing the term and the general term.
Tell if a sequence is arithmetic from a given list.
Tell if a sequence is geometric from a given list.
Find out the general term of an AS.
Find out the general term of an GS.
Sum several terms of an AS.
Sum several terms of a GS.
Sum the infinite terms of a GS (ratio < 1).

Links

OEIS. The Online Enciclopedia of Integer Sequences.
Aquiles and the tortoise.
Zeno's paradox (Numberphile).
Martin Gardner tweet.
Nature by numbers.
Do vampires exist today? (Futility Closet).
¿De dónde viene el número e? (Derivando)

Criterios de evaluación

3.1. Investigar y comprobar conjeturas sencillas tanto en situaciones del mundo real como abstractas de forma autónoma, trabajando de forma individual o colectiva la utilización del razonamiento inductivo y deductivo para formular argumentos matemáticos, analizando patrones, propiedades y relaciones, examinando su validez y reformulándolas para obtener nuevas conjeturas susceptibles de ser puestas a prueba.

4.1. Reconocer patrones en la resolución de problemas complejos, plantear procedimientos, organizar datos, utilizando la abstracción para identificarlos aspectos más relevantes y descomponer un problema en partes más simples facilitando su interpretación computacional y relacionando los aspectos fundamentales de la informática con las necesidades del alumnado.

Refuerzo

Quizizz.