Topicmf6d003ae589539ab_1528449000663_0Topic

Arithmetic Sequence

Levelmf6d003ae589539ab_1528449084556_0Level

Third

Core curriculummf6d003ae589539ab_1528449076687_0Core curriculum

VI. Sequences. Student:

4) checks if a given sequencesequencesequence is arithmetic or geometric.

5) applies the formula for the nth term and for the sum of n first terms of the arithmetic sequencearithmetic sequencearithmetic sequence.

Timingmf6d003ae589539ab_1528449068082_0Timing

45 minutes

General objectivemf6d003ae589539ab_1528449523725_0General objective

To interpret and correctly use the information presented both in the mathematics and general knowledge texts, as well as in the form of diagrams, graphs and tables.

Specific objectivesmf6d003ae589539ab_1528449552113_0Specific objectives

1. To develop English language skills; to develop mathematics, IT, general knowledge and science competences; to develop and master the ability of learning and self‑development.

2. To find terms of the sequencesequencesequence on the basis of general formula.

3. To find the general formulageneral formulageneral formula on the basis of terms of the sequencesequencesequence.

Learning outcomesmf6d003ae589539ab_1528450430307_0Learning outcomes

The student:

- can identify and define terms of the sequencesequencesequence on the basis of the general formulageneral formulageneral formula,

- can find general formulageneral formulageneral formula on the basis of terms of the sequence.

Methodsmf6d003ae589539ab_1528449534267_0Methods

1. Station teaching method.

2. Case studies.

Forms of workmf6d003ae589539ab_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmf6d003ae589539ab_1528450127855_0Introduction

Work in groups. Students solve tasks to review their knowledge of mathematical sequences.

Station 1 – Given a sequencesequencesequence, whose initial terms are: $\frac{1}{4},\frac{1}{16},\frac{1}{36},\frac{1}{64},\frac{1}{100},...$ define a general formulageneral formulageneral formula of this sequencesequencesequence.

Station 2 – Write four initial elements of $\left(a_n\right)$sequencesequencesequence, defined by the following formula $a_n=2-n^2,n\in N_{+}.$

Station 3 – Which of the terms of $\left(a_n\right)$ sequence, defined by $a_n=(n^2-2)(n^2-4)(n-5)$ formula equal zero?

Teacher provides feedback, answers questions and clarifies students’ doubts.

Proceduremf6d003ae589539ab_1528446435040_0Procedure

Teacher informs students about lesson objectives which are: to introduce one of the specific mathematical sequences i.e. arithmetic sequencearithmetic sequencearithmetic sequence.

Students, working individually, solve tasks and formulate conclusions.

Task 1
Find 5 consecutive terms of $\left(a_n\right)$ sequencesequencesequence, described by a general formulageneral formulageneral formula $a_n=-2n+1$. Calculate a common difference(common) differencedifference between the consecutive elements of the sequencesequencesequence. Note down your observations.

Conclusions:

The difference between consecutive terms of the sequence is a constant number.

Teacher informs students, that a number sequencenumber sequencenumber sequence having this property is called an arithmetic sequencearithmetic sequencearithmetic sequence.

Definition

An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
If $\left(a_n\right)$ sequence is an arithmetic sequencesequencesequence with r difference(common) differencedifference, so according to this definition we get $a_{n+1}-a_n=r$.

Students learn a formula for the general term of the arithmetic sequencearithmetic sequencearithmetic sequence.

Let's assume that $\left(a_n\right)$ is an arithmetic sequencesequencesequence with a difference(common) differencedifference of r. According to the definition of the arithmetic sequencearithmetic sequencearithmetic sequence we obtain:

$a_2=a_1+r$

$a_3=a_2+r=(a_1+r)+r=a_1+2r$

$a_4=a_3+r=(a_1+2r)+r=a_1+3r$

$a_5=a_4+r=(a_1+3r)+r=a_1+4r$

$a_n=a_{n-1}+r=[a_1+(n-2\left)r\right]+r=a_1+(n-1)r$

Hence:

$a_n=a_1+(n-1)r$

[Interactive illustration]

Theorem
If $\left(a_n\right)$ sequencesequencesequence is an arithmetic sequencearithmetic sequencearithmetic sequence with a difference of $r$, then according to this definition each $n\in N_{+},a_n=a_1+(n-1)r.$

Students solve tasks individually using the information learnt.

Task 2
Find an r difference of the arithmetic sequence $\left(a_n\right)$, where $a_1=-12,a_{34}=65$.mf6d003ae589539ab_1527752263647_0Find an r difference of the arithmetic sequence $\left(a_n\right)$, where $a_1=-12,a_{34}=65$.

Task 3
Find the first term of the arithmetic sequencearithmetic sequencearithmetic sequence $\left(a_n\right)$, where  $a_7=37,r=5,5$.

Task 4
For which value of k, numbers $a_1=3,a_2=k+1,a_3=3k-6$ are the consecutive terms of the $\left(a_n\right)$ arithmetic sequencearithmetic sequencearithmetic sequence?

Task 5
Three numbers form an arithmetic sequencearithmetic sequencearithmetic sequence. The sum of these numbers equals 20, and the sum of the squares of the first and last numbers equals 218. Find these numbers.

An extra task
In a given arithmetic 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 of this sequence.mf6d003ae589539ab_1527752256679_0In a given arithmetic 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 of this sequence.

Lesson summarymf6d003ae589539ab_1528450119332_0Lesson summary

Students do the revision exercises and formulate conclusions to remember.

- An arithmetic sequencearithmetic sequencearithmetic sequence is a sequencesequencesequence in which the difference(common) differencedifference between a given term of the sequence and the term preceding it, is constant.

- If a sequence $\left(a_n\right)$ is an arithmetic sequencearithmetic sequencearithmetic sequence with the difference(common) differencedifference r, then on the basis of the definition we get $a_{n+1}-a_n=r.$

Selected words and expressions used in the lesson plan

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

arithmetic sequencearithmetic sequencearithmetic sequence

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

general formulageneral formulageneral formula

general term (element)general term (element)general term (element)

number sequencenumber sequencenumber sequence

sequencesequencesequence

terms (elements) of the sequenceterms (elements) of the sequenceterms (elements) of the sequence