The sum of terms of geometric sequence

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Suma wyrazów ciągu geometrycznego

Learning objectives

You will learn the formula for the sum of n initial terms of the geometric sequence.
You will learn how to use the formula for the sum of n initial terms of the geometric sequence.

Learning effect

You will learn the formula for the sum of n initial terms of the geometric sequence,
You will learn to use the formula for the sum of n initial terms of the geometric sequence.

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nagranie abstraktu

Solve the preliminary tasks. They will help you remember the information about geometric sequencesGeometric sequencegeometric sequences you have learned so far.

Task 1

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nagranie abstraktu

Calculate the first five terms of the geometric sequenceGeometric sequencegeometric sequence (aIndeks dolny n) , where aIndeks dolny 1 = 2, q = 3.

Task 2

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nagranie abstraktu

Determine the geometric sequenceGeometric sequencegeometric sequence (aIndeks dolny n), knowing that aIndeks dolny 9= 3, aIndeks dolny 4 = - 3.

Task 3

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nagranie abstraktu

Determine the positive number x so that the numbers 10, x, and 2,5 form a geometric sequenceGeometric sequencegeometric sequence.

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nagranie abstraktu

The aim of the lesson is to determine the formula for calculating the sum of n initial terms of the geometric sequencethe sum of n initial terms of the geometric sequence (aIndeks dolny n).

Watch and analyze the SLIDESHOW showing the way of determining the formula for the sum of n initial terms of the geometric sequenceGeometric sequencegeometric sequence. Formulate the appropriate conclusions.

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nagranie abstraktu

Conclusion:

The sum of the n initial terms of the geometric sequenceGeometric sequencegeometric sequence ( $a_{n}$ ) with the quotient $q \neq 1$ is described by the formula.

S_{n} = a_{1} \frac{1 - q^{n}}{1 - q}

If $q = 1$ , then $S_{n} = n \cdot a_{1}$ .

Using the known formula, solve the tasks.

Task 4

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nagranie abstraktu

The sum of the eight terms of the geometric sequenceGeometric sequencegeometric sequence (aIndeks dolny n) equals to 21675. Knowing that the quotient q is equal to 2, designate the first term aIndeks dolny 1.

Task 5

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nagranie abstraktu

The sum of all terms of the finite geometric sequenceGeometric sequencegeometric sequence (aIndeks dolny 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?

Task 6

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nagranie abstraktu

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

Task 7

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nagranie abstraktu

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

aIndeks dolny 1= $\frac{1}{4}$

Task 8

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nagranie abstraktu

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.

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nagranie abstraktu

On completion of all tasks do the revision exercises.

Remember:

The sum of the n initial terms of the geometric sequenceGeometric sequencegeometric sequence (aIndeks dolny n) with the quotient $q \neq 1$ is described by the formula $S_{n} = a_{1} \frac{1 - q^{n}}{1 - q}$ . If $q = 1$ , the $S_{n} = n \cdot a_{1}$ .

Exercises

Exercise 1

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Wersja alternatywna ćwiczenia: Determine which sentence is true. Możliwe odpowiedzi: 1. The sum of eight initial words of the geometric sequence

(a_{n})

described by the formula

a_{n} = 8 \cdot (\frac{1}{2})^{n}

is equal to 8160., 2. The sum of ten initial terms of the geometric sequence

(a_{n})

in which

a_{3} = 4

and

a_{6} = -32

is equal to 341., 3. The fifth term of the geometric sequence

(a_{n})

, where

s_{2} = 4

and

s_{3} = \frac{13}{3}

is equal to

\frac{1}{27}

., 4. In a geometric sequence

(a_{n})

the sum of n initial terms is described by the formula

s_{n} = 2^{n + 1} - 2

. The seventh term of this sequence is equal to 254.

Determine which sentence is true.

The sum of eight initial words of the geometric sequence $(a_{n})$ described by the formula $a_{n} = 8 \cdot (\frac{1}{2})^{n}$ is equal to 8160.
The sum of ten initial terms of the geometric sequence $(a_{n})$ in which $a_{3} = 4$ and $a_{6} = -32$ is equal to 341.
The fifth term of the geometric sequence $(a_{n})$ , where $s_{2} = 4$ and $s_{3} = \frac{13}{3}$ is equal to $\frac{1}{27}$ .
In a geometric sequence $(a_{n})$ the sum of n initial terms is described by the formula $s_{n} = 2^{n + 1} - 2$ . The seventh term of this sequence is equal to 254.

zadanie

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Exercise 2

Milling machine has 17 spindle speeds. These speeds are continuously geometric. Minimum speed is 31.5 rpm, the largest (the seventeenth) 1134 rpm. Calculate the ninth spindle speed.

Exercise 3

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Indicate which pairs of expressions or words are translated correctly.

ciąg liczbowy - number sequence
ciąg geometryczny - geometric sequence
suma n początkowych wyrazów ciągu geometrycznego - sum of n initial terms of the geometric sequence
wyraz ciągu - term (element) of the sequence
iloraz ciągu - general term of the sequence
kolejne wyrazy ciągu geometrycznego - first(initial) term of the sequence

zadanie

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Match Polish terms with their English equivalents.

ciąg liczbowy
quotient of the sequence
ciąg geometryczny
geometric sequence
iloraz ciągu
Term of the sequence
numerical sequence
wyraz ciągu

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Glossary

consecutive terms of the sequence

kolejne wyrazy ciągu geometrycznego

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wymowa w języku angielskim: consecutive terms of the sequence

first (initial) term of the sequence

pierwszy wyraz ciągu

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

general term of the sequence

wyraz ogólny ciągu geometrycznego

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wymowa w języku angielskim: general term of the sequence

geometric sequence

ciąg geometryczny

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wymowa w języku angielskim: geometric sequence

number sequence

ciąg liczbowy

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wymowa w języku angielskim: number sequence

quotient of the sequence

iloraz ciągu

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wymowa w języku angielskim: quotient of the sequence

term (element) of the sequence

wyraz ciągu

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

the sum of n initial terms of the geometric sequence

suma n początkowych wyrazów ciągu geometrycznego

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wymowa w języku angielskim: the sum of n initial terms of the geometric sequence

Keywords

number sequencenumber sequence number sequence

geometric sequenceGeometric sequencegeometric sequence

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

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

quotient of the sequenceQuotient of the sequencequotient of the sequence

Scenariusz

Exercises

Glossary

Keywords