Arithmetic sequence

R1SI3r42jhkfA

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

RMGl4nI7OLdop1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

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

R1Iqvt1Kv8Ju9

OBIEKT MULTIMEDIALNY KLIKNIJ, ABY OTWORZYĆ NA PEŁNYM EKRANIE

Kliknij, aby rozpocząć

Wystąpił błąd

Spróbuj ponownie później

Ilustracja interaktywna

Trwa wczytywanie danych...

General formula of the arithmetic sequence

Source: GroMar, licencja: CC BY 3.0.

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

R1I9nxxURSbeF1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

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

RI2OvhjWNbyC41

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

Task 5

RuviEeP8qs1901

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

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

RBTdkosCJYHGj1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

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

R8RedFeDzVgBy1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

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

RYmlohPb08kw9

OBIEKT MULTIMEDIALNY KLIKNIJ, ABY OTWORZYĆ NA PEŁNYM EKRANIE

Kliknij, aby rozpocząć

Wystąpił błąd

Spróbuj ponownie później

Determine which sentences are true.

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

zadanie

Source: GroMar, licencja: CC BY 3.0.

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.