Grade 10 Math Q1 Ep7: Finding the nTH term of a Geometric Sequence and Geometric Means

Summary

In today's episode of 'AdaptTV', Sir Jason Flores, the math buddy, guides viewers through the intricacies of geometric sequences. The lesson focuses on identifying terms, calculating the nth term using the formula a_n = a_1 × r^(n-1), and determining geometric means. With engaging examples, viewers learn to find terms and means in sequences, enhancing their logical reasoning and critical thinking skills. The episode concludes with a motivational note, encouraging continuous learning and a love for math.

Takeaways

• 📘 The lesson focuses on enhancing logical reasoning and critical thinking skills through understanding geometric sequences.
• 🔢 It introduces the formula for finding the nth term of a geometric sequence: a_n = a_1 × r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.
• 👨‍🏫 Sir Jason Flores, the presenter, guides viewers through practical examples to apply the formula and find terms of geometric sequences.
• 📈 The script demonstrates how to calculate the seventh and tenth terms of a sequence with a common ratio of 2, using the formula.
• 🧮 An example is provided to find the seventh term of a sequence when the fourth term and the common ratio are known.
• 🔍 The concept of geometric means is explained, which are the terms that lie between the first and last terms (extremes) in a geometric sequence.
• 📐 The method to find geometric means is shown through examples, using the relationship between terms and the common ratio.
• 📝 The lesson includes interactive activities for viewers to practice finding terms and geometric means in sequences, reinforcing learning through engagement.
• 🎓 The script concludes with a summary of the key learnings, emphasizing the importance of perseverance and a positive attitude towards learning math.
• 📺 The lesson is part of the 'AdaptTV' series, encouraging viewers to engage with the content and follow the channel for more educational content.

Q & A

What is the main focus of the lesson presented by Sir Jason Flores in the 'adapttv' episode?

The main focus of the lesson is to help viewers develop their logical reasoning and critical thinking skills by teaching them how to find terms of a geometric sequence, including the nth term and geometric means.

What is the formula used to find the nth term of a geometric sequence?

The formula used to find the nth term of a geometric sequence is a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the number of terms.

How does the video demonstrate finding the seventh term of a geometric sequence with a common ratio of 2?

The video demonstrates finding the seventh term by first determining the common ratio of 2, then calculating the next terms (40, 80, 160), and finally identifying the seventh term as 320 by multiplying the sixth term (160) by the common ratio (2).

What is the 10th term of the geometric sequence starting with 5, with a common ratio of 2?

The 10th term of the geometric sequence starting with 5 and a common ratio of 2 is 2560, calculated using the formula a_10 = 5 * 2^(10-1) = 5 * 2^9 = 5 * 512 = 2560.

How does the video explain the concept of geometric means in a sequence?

The video explains that geometric means are the terms that lie between the first and last terms (extremes) of a geometric sequence, and they can be found using the formula for the nth term of a geometric sequence.

What is the method to find the geometric mean when given the first and last terms of a geometric sequence?

To find the geometric mean when given the first and last terms, use the formula a_2 = sqrt(a_1 * a_3), where a_1 is the first term and a_3 is the last term.

Can you provide an example of how the video finds the geometric mean between the terms 12 and 3?

The video finds the geometric mean between 12 and 3 by setting up the equation a_2^2 = a_1 * a_3 and solving for a_2, where a_1 = 12 and a_3 = 3. The geometric mean a_2 is found to be 6.

How does the video solve for the geometric means in the sequence 2, blank, blank, 250?

The video first determines the common ratio by using the formula r = sqrt[n-k](a_n / a_k) and then multiplies the first term by the common ratio to find the succeeding terms, resulting in the sequence 2, 10, 50, 250.

What is the significance of the formula a_n = a_1 * r^(n-1) in the context of the video?

The formula a_n = a_1 * r^(n-1) is significant as it provides a direct method to calculate any term in a geometric sequence, which is a key concept taught in the video to enhance understanding of geometric sequences.

How does the video conclude the lesson on geometric sequences?

The video concludes by summarizing the key learnings, encouraging viewers to continue their mathematical journey, and reminding them that learning math can be fun and easy, while also promoting the 'adapttv' YouTube channel.

Outlines

00:00

📚 Introduction to Geometric Sequences

Sir Jason Flores, the math buddy, welcomes viewers to an episode of adaptTV focused on enhancing logical reasoning and critical thinking skills. The lesson aims to familiarize viewers with formulas for finding terms in a geometric sequence, calculating the nth term, and determining geometric means. The episode builds on previous knowledge of geometric sequences and introduces the formula for finding the nth term: a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number. An example sequence (5, 10, 20, ...) is used to illustrate finding the 7th term and the 10th term, with the common ratio identified as 2. The 10th term is calculated as 2560 using the formula.

05:05

🔍 Solving for Terms in Geometric Sequences

The second paragraph delves into solving for the seventh term of a geometric sequence where the fourth term is 128 and the common ratio is 4. It guides viewers to first determine the first term using the given fourth term and then apply the formula to find the seventh term. The solution process involves substituting the known values into the formula, solving for the first term, and then using it to find the seventh term, which is calculated to be 8192. The paragraph reinforces the utility of the formula in swiftly determining terms within a geometric sequence.

10:07

🧩 Finding Geometric Means

This section introduces the concept of geometric means, which are terms that lie between the first and last terms (extremes) of a geometric sequence. The lesson demonstrates how to find the geometric mean using the formula and an example sequence with given first and last terms. The process involves substituting values into the formula, performing cross-multiplication, and solving for the geometric mean. The example provided finds the second term of the sequence as six. The paragraph emphasizes the importance of understanding geometric means for solving problems involving sequences.

15:07

🎓 Applying Formulas to Find Geometric Means

The fourth paragraph continues the exploration of geometric means with an example involving a sequence with four terms, where the first and last terms are given. It explains how to determine the common ratio and then use it to find the missing terms, which are the geometric means. The process includes identifying the extremes, calculating the common ratio using the formula r = (a_n / a_k)^(1/(n-k)), and then applying this ratio to find the intermediate terms. The example concludes with the sequence being 2, 10, 50, and 250, showcasing the application of geometric sequence concepts.

20:08

📘 Summary of Geometric Sequence Concepts

The final paragraph summarizes the key learnings from the episode, which include understanding how to find terms in a geometric sequence, calculating the nth term, and determining geometric means. It reiterates the importance of the formula for finding terms and provides a brief overview of the examples covered. The lesson concludes with an encouragement to continue learning and loving math, highlighting the fun and easy approach to understanding geometric sequences.

Keywords

💡Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the video, the concept is central as it is used to explain how to find terms in a sequence, such as the seventh term of the sequence 5, 10, 20, which is found by multiplying the sixth term by the common ratio of 2.

💡Common Ratio

The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. The video script uses the common ratio to calculate further terms in a sequence, exemplified by the sequence 5, 10, 20, where the common ratio is 2, indicating each term is twice the previous one.

💡nth Term

The nth term in a sequence refers to the term in a specific position in the sequence. The video discusses finding the nth term of a geometric sequence using the formula a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, and r is the common ratio. This is illustrated when finding the 10th term of the sequence 5, 10, 20.

💡Geometric Mean

The geometric mean is a term in a geometric sequence that lies between two other terms, with the property that it is the square root of the product of the terms it lies between. The video explains how to find geometric means, using the example of finding the term between 2 and 16, which is 4, since sqrt(2 * 16) = 4.

💡Logical Reasoning

Logical reasoning is the process of thinking through a problem using valid logical principles. The video aims to develop viewers' logical reasoning skills by guiding them through the process of finding terms in a geometric sequence, which requires understanding and applying mathematical rules and formulas.

💡Critical Thinking

Critical thinking involves analyzing and evaluating information to form judgments. The video encourages critical thinking by presenting problems that require viewers to apply mathematical concepts and formulas to find terms in geometric sequences, fostering analytical and evaluative skills.

💡Self-learning Module

A self-learning module is a set of instructional materials designed for independent study. The video serves as a self-learning module for viewers to enhance their understanding of geometric sequences, as indicated by the prompt to prepare a self-learning module, pen, and paper.

💡Formula

In mathematics, a formula is a concise way of expressing information symbolically. The video emphasizes the importance of formulas in finding the nth term and geometric means of a sequence, providing the formula a_n = a_1 * r^(n-1) as a tool for solving geometric sequence problems.

💡Sequence

A sequence is an ordered list of objects or numbers. The video's theme revolves around sequences, specifically geometric sequences, where the order and pattern of numbers are crucial for finding specific terms or understanding the sequence's properties.

💡Terms

In the context of the video, terms refer to the individual numbers in a sequence. The script guides viewers on how to find specific terms within a geometric sequence, such as the seventh term or the tenth term, using mathematical formulas and the properties of the sequence.

💡Ratio

A ratio is a quantitative relationship between two numbers, indicating how many times one number contains or is contained within the other. The video uses the term 'ratio' to describe the relationship between consecutive terms in a geometric sequence, which is essential for finding the common ratio and subsequent terms.

Highlights

Introduction to the lesson on geometric sequences with Sir Jason Flores.

Emphasis on developing logical reasoning and critical thinking skills.

Guidance on preparing self-learning modules, pen, and paper for the lesson.

Explanation of how to find terms of a geometric sequence.

Discussion on determining the geometric mean of a sequence.

Example problem: Finding the seventh term of a geometric sequence.

Demonstration of calculating the common ratio in a sequence.

Formula introduction for finding the nth term of a geometric sequence.

Practical application of the formula to find the 10th term of a sequence.

Activity: Solving for the seventh term with a given fourth term and common ratio.

Step-by-step solution to find the first term using the fourth term.

Calculation of the seventh term using the derived first term.

Introduction to the concept of geometric means in sequences.

Example problem: Finding the geometric mean between given extremes.

Methodology for calculating the geometric mean using cross-multiplication.

Example problem: Inserting geometric means in a sequence with given extremes.

Explanation of how to find the common ratio using the extremes.

Conclusion summarizing the key learnings from the episode.

Encouragement to continue learning and engaging with math.

Transcripts

[Music]

hi

good day welcome in today's episode of

adapttv

i am sir jason flores also your math

buddy

and i will be here to help you in

developing

your logical reasoning and critical

thinking skills

is your self learning module ready

what about your pen and paper

great let's begin a fun and

exciting lesson for this lesson

you are expected to familiarize yourself

with the formulas in finding terms

of geometric sequence

also find the nth term of

a geometric sequence and

determine the geometric mean or

geometric means

of a geometric sequence

episode you learned about geometric

sequences

and how to find the next terms

of geometric sequences

in this episode we will discuss ways in

finding the nth term of a geometric

sequence

for example what is the seventh

term of the sequence 5 10

20 and so on

first let's determine the common ratio

what do you think the common ratio is

that's correct the common ratio of this

sequence

is 2 thus the next

three terms are 40 80

and 160 and you can easily identify

the seventh term when you multiply

160 by two you will obtain

the 7th term which is 320.

now what do you think is the 10th

term using the geometric sequence

5 10 20 and so on

you are asked to find for the 10th term

let us now use a formula which may help

us find

an unknown term of a geometric sequence

the formula in finding the nth term

of a geometric sequence is a sub

n is equal to a sub 1 times

r raised to n minus 1.

again the formula in finding the

nth term of a geometric sequence

is a sub n is equal to a sub 1

times r raised to n minus 1

wherein a sub n is the nth term

a sub 1 is the first term

r is the common ratio and

n is the number of terms

using the sequence 5 10

20 and so on let's use

the formula in finding the tenth term of

the geometric sequence given

a sub one or the first term is five

the ratio which is equal to two n

is ten again we are looking

for the 10th term now let's

substitute the values in the formula

our a sub n will be a sub 10

equal to our a sub 1 is 5

our r is 2

raised to n our n again is 10 so that's

10

minus 1. next we'll have a sub 10

is equal to 5 times

2 raised to 10 minus 1

will give us 9. then you have a sub 10

is equal to five times

two raised to nine it means that you

have to multiply two

by itself nine times will give us

five hundred twelve

a sub 10 now is the product

of 5 and 512 will give us

that's right 2560.

thus the tenth term

of the geometric sequence is 2560.

incredible now let's have another

activity

what is the seventh term of a geometric

sequence

whose fourth term is 128

and the common ratio is equal to four

to begin with the problem you must have

to analyze

carefully what does it ask for

the problem is asking for the seventh

term

but the first term was not given

first identify the given values and

the unknown variables for this problem

the given terms are a sub 4 or the

fourth term which is equal to 128

and the common ratio which is equal to

four

there are two unknowns the first term or

a sub one

and the seventh term or a sub 7.

now let's use the formula a sub n

is equal to a sub 1 times r raised to n

minus 1. again there are two unknowns

in the problem and to solve for a sub 7

we need to solve first for a sub 1.

since the given term is the fourth term

which is equal to 128

we can use it to solve for the value of

a sub 1.

now let's substitute the value of a sub

4

which is equal to 128

and which is 4 and r

which is also 4 in the formula a sub

n is equal to a sub 1 times r

raised to n minus 1. we will have 128

is equal to a sub 1

times the ratio which is 4

our n is also 4

minus 1. then

you have 128 is equal to

let's copy first a sub 1 times 4

and 4 minus 1 will give us 3.

next we will have 128

is equal to a sub one

times four multiplying four by itself

three times will give us 64.

then divide both sides

by 64 so we can isolate a sub 1.

64 divided by 64 will give us one what's

left is a sub 1

and 128 divided by 64

will give us 2 thus the first term of

the sequence

is equal to 2.

since we have our first term we can now

solve

for the unknown term which is a sub 7.

again

using the formula a sub n is equal to a

sub 1

times r raised to n minus 1.

that's a sub 7 is equal to

a sub 1 which is 2

times r which is four

raised to seven minus one

next we will have a sub seven

is equal to two times 4

raised to 7 minus 1 will give us 6.

a sub 7 is equal to 2

times we will multiply 4 by itself

6 times will give us 4

0096 and finally

multiplying 2 to 4096

will give us 8192

thus the first term

and the seventh term of the sequence is

2 and 8192

respectively great job indeed

now it's more swift and convenient to

find

the nth term of a geometric sequence

using the formula right

now let's try to see what you have

learned from today's episode

by answering this question

find the specified term of the geometric

sequence

given the first term and the common

ratio

the problem tells us to solve for the

fifth

term before that

let's determine first the given values

we have a sub 1 which is equal to 3

and the common ratio which is equal to

3.

after that let's use the formula in

finding the nth

term of a geometric sequence

a sub n is equal to a sub 1

times r raised to n minus 1.

we have a sub 5 is equal to

our a sub 1 or the first term is 3

times the common ratio which is also 3

our n is 5 minus one

then you will have a sub five is equal

to three

times three raised to five

minus one is correct

four a sub five

is equal to three then multiply

three by itself four times

that's three times three times three

times three will give us

correct eighty-one a sub five is equal

to the product of three

and eighty-one will give us two hundred

forty-three

thus the fifth term of the sequence

is equal to 243

also awesome well how was the experience

so far

wonderful to further enhance your

knowledge and skills about this topic

let us discover a shorter way to

identify

the unknown term or terms in between

terms of geometric sequences

also known as geometric means

let's take a look at this example

twelve blank

three given terms

are first and last terms

these terms are called the extremes

and the term or terms in between the

extremes

are called geometric mean or geometric

means

in the sequence 2

4 8 16

the numbers 4 and 8 are the geometric

means

of the extremes 2 and

16. i know that you are very eager to

explore more about this lesson

well let's begin

let's go back to the problem 12

blank 3

the first term is 12 and

the last term is 3.

now let us substitute but remember

the common ratio refers to the ratio of

two consecutive terms with that

we will use a sub 3

divided by a sub 2

is equal to a sub 2

divided by a sub 1.

now let's replace it with the given

values

for a sub 3 or the third term we have 3

over or divided by the second term which

is a sub 2

is equal to a sub 2 is still unknown so

we'll just write a sub 2

divided by the first term a sub 1 which

is

12. next let's do

cross multiplication for this part we

will multiply

a sub 2

times a sub 2

equal to 12

times 3

a sub 2 times a sub 2 will give us

a sub 2 squared that's correct and

12 times 3 will give us

correct that's 36.

next let's apply getting the square of

each terms we'll have

the square root of this a sub 2 squared

what's left will be a sub 2

and the square root of 36

is six

thus the geometric mean or the second

term

of the geometric sequence is six

buckle up fasten your seat belts as we

move on to the next example

come and join me as we solve this

problem together

in the geometric sequence 2 blank

blank 250

there are two geometric means needed

in this problem let us identify first

the extremes and the number of terms

the extremes are 2 and

250 and there are

four terms in the sequence

so to insert terms let us identify first

the common ratio by using the formula

r is equal to n minus k

and the square of a sub n

all over a sub k with a given

a sub 1 which is equal to 2 a sub 4

which is equal to 250 our n

which is equal to 4 and k which is equal

to 1. now let's substitute the given

values

to the formula we will have

r is equal to

our n again is four minus

1 and the square of

our a sub 4

divided by our a sub 1.

now use the values we have r

is equal to 4 minus 1. our a sub 4 again

is equal to 250

over a sub 1 which is 2.

then we will have r is equal to 4 minus

1 will give us 3

and the square of 250

divided by 2.

next we will have r

is equal to the cube root of

250 divided by 2 will give us

125 getting the cube root

of 125 that is equal to

five correct

now the common ratio of this geometric

sequence

is equal to five

to get the succeeding term remember

to multiply the preceding term

times the common ratio

since we have 2 as our first term

we will multiply it by the common ratio

which is

five so two times five

will give us ten

and to get the third term we will also

multiply

the second term by the common ratio

which is 5

so 10 times 5 will give us

50.

therefore the sequence is

2 10 50

and 250.

congratulations that was very nice

thanks for helping me out

i hope you already developed the

knowledge and skills you need

in finding geometric means

keep it up to keep the fire burning

join me once more as we solve the

following problems together

for letter a find the geometric mean

of the given extremes three blank

and eight again we are looking

for the geometric mean of the extremes

three and eight with that

let's use a sub three

divided by a sub 2

is equal to a sub 2 divided

by a sub 1. let's substitute the values

our a sub 3 or the third term is eight

divided by the second term is still

unknown

so we will just copy a sub 2

is equal to a sub 2 again is unknown

just copy

divided by our a sub 1 or the first term

which is equal to 3. now

let's cross multiply these terms you

have a sub 2

times a sub 2.

is equal to three times

eight all right

multiplying a sub two times a sub two

will give

us okay that's correct

a sub 2 squared

and 3 times 8 will give us

all right that's 24.

next let's get the square of both sides

what's left here is

a sub 2. notice that

24 is not a perfect square so we will

look

for two factors

of 24 which is four

times six

then we will have a sub two

since four is a perfect square the

square root of four

is correct

that's two square root of

six

so the geometric mean of the sequence

is two square root of six

now let's proceed to letter b

insert geometric means in the geometric

sequence

to blank blank

686

again we are looking for two unknowns

which are a sub 2 and a sub 3.

with that we will use the formula first

in getting the ratio which

is square root of a sub n

divided by a sub k and n minus

k let's

substitute the given values

so r is equal to

our n again is four

minus one and the square root of a sub

n we will use the value a sub 4

and our a sub k is a sub

1. next we will have

r is equal to 4 minus 1

and the square of a sub 4 our fourth

term is 686

divided by our a sub 1 or the first term

which is

2. moving on we have r

is equal to 4 minus 1 is 3

and 686 divided by 2 will give us

343

getting the cube root of 343

will give us correct

that's seven

since you already have our common ratio

we can now solve

and find the missing terms

again that is a sub 2 and a sub 3.

to get a sub two we will multiply

the first term by the common ratio

seven so that's two

times 7 will give us

14 correct

and to get the third term we will

multiply

the second term 14 times

the ratio which is seven

so fourteen

times seven will give

us

98

so the geometric means of the sequence

are 14 and 98.

to sum it up in today's episode you're

able to familiarize yourself

with the formula in finding terms of

geometric sequence

also you were able to find the nth

term of a geometric sequence

and determine the geometric mean

or means of a geometric sequence

awesome i hope you learned a lot

never give up and remember winners

never quit and quitters never win

keep loving math and that concludes

our lesson for today see you again on

the next episode

and please don't forget to like share

and subscribe to the deaf tv

official youtube channel and this has

been sir

jason flores also bear in mind

that learning math will always be fun

and easy be awesome be

awesome only here on deputy

[Music]

you