Properties of arithmetic sequences

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Własności ciągów arytmetycznych

Learning objectives

You will learn the properties of the arithmetic sequence

You will get to know the relationship between the 3 consecutive terms of the sequence

Learning effect

You will learn the properties of the arithmetic sequence
You will use the relationship between 3 consecutive terms of the arithmetic sequence to define the properties of arithmetic sequence

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nagranie abstraktu

Organize your knowledge about numerical sequences, the ways how to describe them, and the knowledge of arithmetic sequencearithmetic sequencearithmetic sequence. The aim of the lesson is to learn about the properties of the arithmetic sequence.

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nagranie abstraktu

Analyse the animation showing the relationship between three consecutive terms of the arithmetic sequence. Formulate the appropriate conclusion.

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nagranie abstraktu

Each term of the arithmetic sequence (except the first and last, if the sequence is finite) is the arithmetic mean of the adjacent terms.

Using available resources find the theorem to prove your conclusions.

arithmetic sequence

Theorem: arithmetic sequence

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nagranie abstraktu

The sequence (aIndeks dolny n) is an arithmetic sequence if, and only if, every term of this sequence except the first (and possibly the last) term is the arithmetic mean of the preceding and the subsequent term.

a_{n} = \frac{a_{n + 1} + a_{n - 1}}{2}

Study the monotonicity of the arithmetic sequencearithmetic sequencearithmetic sequence.

Formulate the appropriate conclusions.

Task 1

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nagranie abstraktu

Draw a graph of the arithmetic sequence (aIndeks dolny n) with the general formula (aIndeks dolny n) Determine the monotonicity of this sequence. Note down an appropriate conclusion.

$a_{n} = 3 n + 4, a_{n} = - \frac{2}{3} n - 3, a_{n} = - 4$

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nagranie abstraktu

The arithmetic sequencearithmetic sequencearithmetic sequence with the difference r is:

an increasing sequenceincreasing sequenceincreasing sequence if the difference r is positive,
a decreasing sequencedecreasing sequencedecreasing sequence if the difference r is negative,
constant if the difference r is equal to zero.

Use the acquired knowledge to solve the following tasks.

Task 2

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nagranie abstraktu

Enter three numbers between numbers 7 and 23, so that, along with the data, they form an arithmetic sequence.

Task 3

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nagranie abstraktu

In arithmetic sequence: (aIndeks dolny n): aIndeks dolny 5 = -31, aIndeks dolny 10 = -66. Calculate aIndeks dolny 1 i r.

Determine the monotonicity of this sequence.

Task 4

An extra task

For which values of x, the numbers 3x + 1, 2x - 4, 5x + 3 (in the order given) form the arithmetic sequence?

On completion of all tasks do the revision tasks.

Remember:

The sequence (aIndeks dolny n) is an arithmetic sequence if, and only if, every term of this sequence except the first (and possibly the last) term is the arithmetic mean of the preceding and the subsequent term,
The arithmetic sequence with the difference r is:
- an increasing sequenceincreasing sequenceincreasing sequenceif the difference r is positive,
- a decreasing sequencedecreasing sequencedecreasing sequence if the difference r is negative,
- constant if the difference r is equal to zero.

Exercises

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Exercise 1

Wersja alternatywna ćwiczenia: Determine which sentences are true. Możliwe odpowiedzi: 1. If we enter number 16 between numbers 8 and 20, we will get three consecutive terms of the arithmetical sequence., 2. If in arithmetic sequence

(a_{n})

with a difference r:

a_{2} = 5

and

a_{22} = 45

then

r = 2

., 3. If in arithmetic sequence

(a_{n})

a_{4} + a_{6} = - 32

, then

a_{5} = 16

., 4. The arithmetic sequence

(a_{n})

described with a general formula

a_{n} = 2 n - 2

a_{n}

increasing sequence.

Determine which sentences are true.

If we enter number 16 between numbers 8 and 20, we will get three consecutive terms of the arithmetical sequence.
If in arithmetic sequence $(a_{n})$ with a difference r: $a_{2} = 5$ and $a_{22} = 45$ then $r = 2$ .
If in arithmetic sequence $(a_{n})$ $a_{4} + a_{6} = - 32$ , then $a_{5} = 16$ .
The arithmetic sequence $(a_{n})$ described with a general formula $a_{n} = 2 n - 2$ is $a_{n}$ increasing sequence.

zadanie

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Exercise 2

Angles of quadrilateral are consecutive terms of the arithmetic sequence.

The smallest angle is 60°. Calculate the measures of the remaining angles of this quadrangle.

Exercise 3

Describe the properties of the arithmetic sequence in English.

Exercise 4

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Indicate which pairs of expressions or words are translated correctly.

ciąg arytmetyczny - arithmetic sequence
różnica - difference
kolejne wyrazy ciągu arytmetycznego - consecutive terms ( elements ) of the arithmetic sequence
ciąg rosnący - increasing sequence
ciąg malejący - constant sequence
ciąg stały - decreasing sequence

zadanie

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Match Polish terms with their English equivalents.

ciąg arytmetyczny
difference
arithmetic sequence
ciąg malejący
increasing sequence
kolejne wyrazy ciągu arytmetycznego
decreasing sequence
ciąg rosnący
różnica
Consecutive terms of the arithmetic sequence

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

arithmetic sequence

ciąg arytmetyczny

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wymowa w języku angielskim: arithmetic sequence

(common) difference

różnica

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wymowa w języku angielskim: (common) difference

consecutive terms (elements) of the arithmetic sequence

kolejne wyrazy ciągu arytmetycznego

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wymowa w języku angielskim: consecutive terms ( elements ) of the arithmetic sequence

increasing sequence

ciąg rosnący

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wymowa w języku angielskim: increasing sequence

decreasing sequence

ciąg malejący

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wymowa w języku angielskim: decreasing sequence

constant sequence

ciąg stały

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wymowa w języku angielskim: constant sequence

Keywords

arithmetic sequencearithmetic sequencearithmetic sequence

(common) difference(common) difference(common) difference

consecutive terms (elements) of the arithmetic sequenceconsecutive terms (elements) of the arithmetic sequence

increasing sequenceincreasing sequenceincreasing sequence

decreasing sequencedecreasing sequencedecreasing sequence

Scenariusz

Exercises

Glossary

Keywords