Topicm9b11e89f97c9fa01_1528449000663_0Topic

The sum of terms of geometric sequence

Levelm9b11e89f97c9fa01_1528449084556_0Level

Third

Core curriculumm9b11e89f97c9fa01_1528449076687_0Core curriculum

VI. Sequences. Student:

1) calculates terms of sequences determined by the explicit formula;

6) applies the formula to the nIndeks górny th^th term and for the sum of n initial terms of the geometric sequence

Timingm9b11e89f97c9fa01_1528449068082_0Timing

45 minutes

General objectivem9b11e89f97c9fa01_1528449523725_0General objective

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

Specific objectivesm9b11e89f97c9fa01_1528449552113_0Specific objectives

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

2. To get to know the formula for the sum of n initial terms of the geometric sequence.

3. To apply the formula of the sum of n initial terms of the geometric sequence.

Learning outcomesm9b11e89f97c9fa01_1528450430307_0Learning outcomes

The student:

- learns the formula for the sum of n initial terms of the geometric sequence,

- applies the formula for the sum of n initial terms of the geometric sequence.

Methodsm9b11e89f97c9fa01_1528449534267_0Methods

1. Task‑based competition.

2. Case study.

Forms of workm9b11e89f97c9fa01_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm9b11e89f97c9fa01_1528450127855_0Introduction

The lesson begins with a short, individual task competition. The first 3 students who solve the tasks correctly will be rewarded with the best grade (5).

Competition.

Task

Calculate the first five terms of the geometric sequence.

(aIndeks dolny n_n), where aIndeks dolny 1₁ = 2, q = 3.

Answer. aIndeks dolny 1₁ = 2, aIndeks dolny 2₂ = 6, aIndeks dolny 3₃ = 18, aIndeks dolny 4₄ = 54, aIndeks dolny 5₅= 162.

Task

Determine the geometric sequence (aIndeks dolny n_n), knowing that aIndeks dolny 9₉ = 3, aIndeks dolny 4₄ = - 3.

Answer. aIndeks dolny 1₁ = 3, q = ( - 1).

Task

Determine the positive number x so that the numbers 10, x, 2.5 form a geometric sequence

Answer. x = 5.

The teacher summarizes the results of the competition and clarifies students’ doubts.

Procedurem9b11e89f97c9fa01_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to determine the formula for calculating the sum of n initial terms of the geometric sequence (aIndeks dolny n_n).

Students watch and analyze SLIDESHOW showing the way of determining the formula for the sum of n initial terms of a geometric sequence. They formulate the appropriate conclusions.

[SLIDESHOW]

Conclusion The sum of the n initial terms of the geometric sequence ($a_n$) with the quotient $q\ne 1$ is described by the formula.

$S_n=a_1\frac{1-q^n}{1-q}$

If $q=1$, then $S_n=n·a_1$.

Students use the formula to solve tasks.

Task

The sum of the eight terms of the geometric sequence (aIndeks dolny n_n) equals to 21675. Knowing that the quotient q is equal to 2, designate the first term aIndeks dolny 1₁.m9b11e89f97c9fa01_1527752263647_0The sum of the eight terms of the geometric sequence (aIndeks dolny n_n) equals to 21675. Knowing that the quotient q is equal to 2, designate the first term aIndeks dolny 1₁.

Answer: aIndeks dolny 1₁ = 85m9b11e89f97c9fa01_1527752263647_0Answer: aIndeks dolny 1₁ = 85.

Task

The sum of all terms of the finite geometric sequence (aIndeks dolny n_n) is equal to 1000. The first term of this sequence is 25, and the quotient equals 3. How many terms does this sequence consist of?m9b11e89f97c9fa01_1527752256679_0The sum of all terms of the finite geometric sequence (aIndeks dolny n_n) is equal to 1000. The first term of this sequence is 25, and the quotient equals 3. How many terms does this sequence consist of?

Answer: n = 4.m9b11e89f97c9fa01_1527752256679_0Answer: n = 4.

Task

The first term of the geometric sequence (aIndeks dolny n_n)with the quotient q = $\sqrt{5}$ equals $\sqrt{5}$ - 1. How many initial terms should you add to receive 124?m9b11e89f97c9fa01_1527712094602_0The first term of the geometric sequence (aIndeks dolny n_n)with the quotient q = $\sqrt{5}$ equals $\sqrt{5}$ - 1. How many initial terms should you add to receive 124?

Answer: n = 6.m9b11e89f97c9fa01_1527712094602_0Answer: n = 6.

Task

Calculate the first term of the geometric sequence (aIndeks dolny n_n) with the quotient q = 3 if the sum of the six initial terms equals 91.

Answer: aIndeks dolny 1₁= $\frac{1}{4}$.

Extra activity

The first segment of the broken line is 80 cm long and each subsequent one is 2 times shorter than the preceding one. How many segments does the broken line consist of, if it is 158.75 cm long?

Answer: n = 7.

Lesson summarym9b11e89f97c9fa01_1528450119332_0Lesson summary

Students do the revision tasks.

Then they summarize the lesson together, formulating conclusions to remember.

The sum of the n initial terms of the geometric sequence ($a_n$) with the quotient $q\ne 1$ is described by the formula.

$S_n=a_1\frac{1-q^n}{1-q}$

If $q=1$, then $S_n=n·a_1$.

Selected words and expressions used in the lesson plan

consecutive terms of the sequenceConsecutive terms of the sequenceconsecutive terms of the sequence

general term of the sequenceGeneral term of the sequencegeneral term of the sequence

geometric sequenceGeometric sequencegeometric sequence

number sequencenumber sequencenumber sequence

quotient of the sequenceQuotient of the sequencequotient of the sequence

term (element) of the sequenceTerm (element) of the sequenceterm (element) of the sequence

the sum of n initial terms of the geometric sequenceThe sum of n initial terms of the geometric sequencethe sum of n initial terms of the geometric sequence