Arithmetic sequence

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Ciąg arytmetyczny

Learning objectives

You will learn the basic concepts of finding terms of sequence on the basis of general formula.

You will learn basic concepts of defining general formula on the basis of the terms of the sequence.

Learning effect

You will find terms of the sequence on the basis of general formula.
You will find the general formula on the basis of terms of the sequence.

Try to brainstorm your knowledge of sequences. Do the task below.

Task 1

Write four initial terms of $(a_{n})$ sequencesequencesequence, defined by the following formula $a_{n} = 2 - n^{2}, n \in N_{+} .$

The main objective of the lesson is to learn about arithmetic sequencearithmetic sequencearithmetic sequence. Do the task and formulate conclusions.

Task 2

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nagranie abstraktu

Find five consecutive terms of $(a_{n})$ sequencesequencesequence, described by a general formulageneral term (element)general formula $a_{n} = - 2 n + 1$ . Calculate a common difference(common) differencedifference between the consecutive terms of the sequencesequencesequence. Note down your observations.

Conclusion:

The difference(common) differencedifference between consecutive terms of the sequencesequencesequence is a constant number.
Using different sources of information read about properties of such sequencesequencesequence.

Arithmetic sequence

Definition: Arithmetic sequence

An arithmetic sequencearithmetic sequencearithmetic sequence is a sequencesequencesequence of numbers such that the difference(common) differencedifference between the consecutive terms is constant.
If $(a_{n})$ sequencesequence sequence is an arithmetic sequencearithmetic sequencearithmetic sequence with r difference(common) differencedifference, then, on the basis of this definition, we get $a_{n + 1} - a_{n} = r$ .

a_{2} = a_{1} + r

a_{3} = a_{2} + r = (a_{1} + r) + r = a_{1} + 2 r

a_{4} = a_{3} + r = (a_{1} + 2 r) + r = a_{1} + 3 r

a_{5} = a_{4} + r = (a_{1} + 3 r) + r = a_{1} + 4 r

a_{n} = a_{n - 1} + r = [a_{1} + (n - 2) r] + r = a_{1} + (n - 1) r

Hence:

a_{n} = a_{1} + (n - 1) r

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General formula of the arithmetic sequence

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n term of the sequence

Theorem: n term of the sequence

If $(a_{n})$ sequencesequencesequence is an arithmetic sequencearithmetic sequencearithmetic sequence with a difference(common) differencedifference of $r$ , then on the basis of this definition each $n \in N_{+}, a_{n} = a_{1} + (n - 1) r$ .

Using the information above do the following tasks.

Task 3

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nagranie abstraktu

Find an $r$ difference(common) differencedifference of the arithmetic sequencearithmetic sequencearithmetic sequence $(a_{n})$ , where $a_{1} = - 12, a_{34} = 65$ .

Task 4

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nagranie abstraktu

Find the first term of the arithmetic sequencearithmetic sequencearithmetic sequence $(a_{n})$ , where $a_{7} = 37, r = 5, 5$ .

Task 5

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nagranie abstraktu

For which value of $k$ , numbers $a_{1} = 3, a_{2} = k + 1, a_{3} = 3 k - 6$ are the consecutive terms of the $(a_{n})$ arithmetic sequencearithmetic sequencearithmetic sequence?

Task 6

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nagranie abstraktu

Three numbers create an arithmetic sequencearithmetic sequencearithmetic sequence. The sum of these numbers equals 20, and the sum of the squares of the first and the last number equals 218. Find these numbers.

Task 7

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nagranie abstraktu

An extra task:
In a given arithmetic sequencearithmetic sequencearithmetic sequence the sum of the 4th and 9th term equals 86, and the sum of the 2nd and 13th term is 22. Find the first term and the common difference(common) differencedifference of this sequencesequencesequence.

On completion of all the tasks do the revision exercises.

An arithmetic sequencearithmetic sequencearithmetic sequence is a sequence in which the difference(common) differencedifference between a given term of the sequencesequencesequence and the term preceding it, is constant for the given sequencesequencesequence.
If a sequence $(a_{n})$ is an arithmetic sequencearithmetic sequencearithmetic sequence with the difference(common) differencedifference $r$ , then on the basis of the definition we get $a_{n + 1} - a_{n} = r .$

Exercises

Exercise 1

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Determine which sentences are true.

$(a_{n})$ sequence of a general term $a_{n} = 3 n^{2} + 4$ is an arithmetic sequence.
The $7^{t h}$ term of the arithmetic sequence $(a_{n}), a_{n} = - \frac{1}{2} (n + 2)$ is a number $(- 4, 5)$ .
If in arithmetic sequence $(a_{n}), a_{5} = 14$ and $a_{10} = 24$ , then $a_{1} = 6$ .
If in arithmetic sequence $(a_{n}), a_{4} = 19$ and $a_{10} = 61$ , then $r = 6$ .

zadanie

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Exercise 2

Let’s assume we analyse an arithmetic sequence $(a_{n})$ with the first term $a_{1} = - 100$ and a common difference $r = 5$ .
Are there any terms of the sequence greater than the number 550? If yes, give at least one such term.

$n > 131, eg . a_{132} = 555$

Exercise 3

Explain in English what an arithmetic sequence is. Use full sentences.

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Exercise 4

Indicate which pairs of expressions or words are translated correctly.

ciąg liczbowy - number sequence
wyraz ogólny - terms (elements) of the sequence
pierwszy wyraz ciągu - the first (initial) term (element) of the sequence
ciąg - sequence
ciąg geometryczny - geometric sequence
wyrazy ciągu - general term (element)

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Match Polish terms with their English equivalents.

ciąg geometryczny
pierwszy wyraz ciągu
sequence
geometric sequence
ciąg liczbowy
ciąg
numeric sequence
the first (initial) element (term) of the sequence
różnica
difference

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Glossary

(common) difference

różnica

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wymowa w języku angielskim: (common) difference

arithmetic sequence

ciąg arytmetyczny

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wymowa w języku angielskim: arithmetic sequence

the first (initial) term (element) of the sequence

pierwszy wyraz ciągu

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wymowa w języku angielskim: the first (initial) term (element) of the sequence

general formula

wzór ogólny

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wymowa w języku angielskim: general formula

general term (element)

Wyraz ogólny

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wymowa w języku angielskim: general term (element)

number sequence

ciąg liczbowy

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wymowa w języku angielskim: number sequence

sequence

ciąg główny

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nagranie abstraktu

terms (elements) of the sequence

Wyrazy ciągu

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

Keywords

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

arithmetic sequencearithmetic sequencearithmetic sequence

number sequencenumber sequencenumber sequence

sequencesequencesequence

the first (initial) element (term) of the sequencethe first (initial) element (term) of the sequence

Scenariusz

Exercises

Glossary

Keywords