Topicm0e2ec77d3f8b6180_1528449000663_0Topic

Arithmetic and geometric sequencegeometric sequencegeometric sequence - applications

Levelm0e2ec77d3f8b6180_1528449084556_0Level

Third

Core curriculumm0e2ec77d3f8b6180_1528449076687_0Core curriculum

VI. Sequences. Student:

7) Uses properties of arithmetic and geometric sequences to solve tasks both in theory and practical applications.

Timingm0e2ec77d3f8b6180_1528449068082_0Timing

45 minutes

General objectivem0e2ec77d3f8b6180_1528449523725_0General objective

To interpret and manipulate information presented in both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesm0e2ec77d3f8b6180_1528449552113_0Specific objectives

1 To communicate in English; develop mathematics and basic scientific, technical and IT competences, shape and develop learning skills.

2 To define the terms of the arithmetic and geometric sequences and certain relationships between these terms.

3 To apply both geometric and arithmetic sequences to solve practical problems.

Learning outcomesm0e2ec77d3f8b6180_1528450430307_0Learning outcomes

The student:

- defines the terms of arithmetic and geometric sequences on the basis of the given relationships and dependences between these terms,

- applies properties of arithmetic and geometric sequences to solve practical problems.

Methodsm0e2ec77d3f8b6180_1528449534267_0Methods

1. Sentence completion.

2. Case study.

Forms of workm0e2ec77d3f8b6180_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm0e2ec77d3f8b6180_1528450127855_0Introduction

Using sentence completion technique, students review the previously learned knowledge of arithmetic and geometric sequences:

- Sequence (aIndeks dolny n_n) is called an arithmetic sequencearithmetic sequencearithmetic sequence if ...

- The arithmetic sequence (aIndeks dolny n_n) is increasing if ...

- To calculate the sum of n initial terms of the arithmetic sequencearithmetic sequencearithmetic sequence, ...

- A geometric sequencegeometric sequencegeometric sequence is a number sequence (aIndeks dolny n_n), in which ...

- The sum of the n initial terms of the geometric sequencegeometric sequencegeometric sequence (aIndeks dolny n_n) with the quotient q ≠ 1 is calculated according to the formula ...

Procedurem0e2ec77d3f8b6180_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to use the previously learned formulas associated with the arithmetic sequences and formulas associated with the geometric sequencegeometric sequencegeometric sequence.

Task
Students, working in groups, watch and analyze the Slideshow, showing the way of solving a task with the use of sequence properties.

[Slideshow]

Conclusions:
- When solving tasks that involve arithmetic and geometric sequences, the properties of both sequences should be used simultaneously.m0e2ec77d3f8b6180_1527752263647_0- When solving tasks that involve arithmetic and geometric sequences, the properties of both sequences should be used simultaneously.

Students work independently and solve tasks.

Task
Between numbers 2 and 30, insert two numbers in such a way that the first three numbers form a geometric sequencegeometric sequencegeometric sequence, and the last three form an arithmetic sequencearithmetic sequencearithmetic sequence.

Answer:
We get two pairs of numbers: ( - 5) and 12,5 or 6 and 18.

Task
The following numbers are presented: 3, x, y, 25. The first three form an increasing arithmetic sequence, the last three - a geometric sequence. Calculate x and y.m0e2ec77d3f8b6180_1527752256679_0The following numbers are presented: 3, x, y, 25. The first three form an increasing arithmetic sequence, the last three - a geometric sequence. Calculate x and y.
Answer:
x = 9, y = 15.

Task
A bacteriologist studying the growth of bacterial cells found that the bacteria on the agar plate doubled their number every 10 minutes. Assuming the initial sample contained 100 bacteria, calculate the number of microorganisms after 4 hours of culture.

Answer:
1677721600 bacteria.

Task
The ball thrown from the balcony of the apartment located on the second floor of the building bounces many times off the concrete ground of the yard. After each consecutive bounce, it loses some energy and rises to $\frac{2}{3}$ of the previous height. Calculate the path in the vertical (up and down) of the ball to the fourth bounce, knowing that the second floor of the building is located 9 meters above the yard’s surface.

Answer:
$37\frac{8}{9}m$

An extra task
The cuboid edge lengths form a geometric sequencegeometric sequencegeometric sequence. The volume of this cuboid is equal to 1000 cmIndeks górny 3³ and the total surface area is 700 cmIndeks górny 2². Calculate the edge lengths of the cuboid.

Answer:
Edge lengths are equal to 5 cm, 10 cm, 20 cm.

Lesson summarym0e2ec77d3f8b6180_1528450119332_0Lesson summary

Students do the revision exercises.

Together, they formulate the rule to remember:

- When solving tasks that involve arithmetic and geometric sequences, the properties of both sequences should be used simultaneously.m0e2ec77d3f8b6180_1527752263647_0- When solving tasks that involve arithmetic and geometric sequences, the properties of both sequences should be used simultaneously.

Selected words and expressions used in the lesson plan

arithmetic sequencearithmetic sequencearithmetic sequence

geometric sequencegeometric sequencegeometric sequence

properties of arithmetic sequenceproperties of arithmetic sequenceproperties of arithmetic sequence

properties of geometric sequenceproperties of geometric sequenceproperties of geometric sequence

quotient of the geometric sequencequotient of the geometric sequencequotient of the geometric sequence

sum of the initial termssum of the initial termssum of the initial terms

sum of the initial terms of arithmetic sequencesum of the initial terms of arithmetic sequencesum of the initial terms of arithmetic sequence

sum of the initial terms of geometric sequencesum of the initial terms of geometric sequencesum of the initial terms of geometric sequence