[TAGALOG] Grade 10 Math Lesson: SOLVING GEOMETRIC SEQUENCE (Part III)- FINDING THE COMMON RATIO

Summary

In this educational video, the host guides viewers through solving for the common ratio in geometric sequences. They use the formula \( a_n = a_1 \cdot r^{(n-1)} \) to find the ratio with examples: one with a first term of 7 and a fifth term of 112, and another with a first term of 5 and a fourth term of 320. The host demonstrates step-by-step calculations, including dividing terms and simplifying expressions to isolate the common ratio. A challenge problem is presented with a first term of 2 and a fourth term of 432, encouraging viewers to apply the learned method. The video concludes with an additional challenge and an invitation to join the next session.

Takeaways

• 📚 The video is a tutorial on finding the common ratio of a geometric sequence.
• 🎓 The formula used to find the common ratio is 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.
• 🔢 The first example involves a sequence with the first term of 7 and the fifth term of 112.
• 🧮 To solve for the common ratio, the presenter divides the fifth term by the first term and then takes the fourth root of the result.
• 📈 The second example features a sequence with the first term of 5 and the fourth term of 320, leading to a common ratio of 4.
• 📉 The presenter emphasizes the importance of correctly substituting the values into the formula and solving for the common ratio.
• 🕵️‍♂️ The video includes a challenge problem for viewers to solve on their own, involving a sequence with the first term of 2 and the fourth term of 432.
• 🏁 The final common ratio for the challenge problem is found to be 6, using the same method as the examples.
• 🎉 The video concludes with an additional challenge question involving a sequence with the first term of 1 and a common ratio of 3, where the term in question is 243.
• 📢 The presenter encourages viewers to share the tutorial with friends, subscribe, and turn on notifications for more educational content.

Q & A

What is the formula for finding the nth term of a geometric sequence?

The formula for the nth term of a geometric sequence is aₙ = a₁ × rⁿ⁻¹, where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the term number.

How can you calculate the common ratio of a geometric sequence given the first and fifth terms?

To find the common ratio, use the formula aₙ = a₁ × rⁿ⁻¹. Substitute the first term and fifth term into the equation, solve for r raised to the fourth power, and then take the fourth root to find r.

What is the common ratio if the first term of a geometric sequence is 7 and the fifth term is 112?

The common ratio is 2. This is calculated by dividing the fifth term (112) by the first term (7) and then taking the fourth root of the result.

How do you solve for the common ratio when given the first and fourth terms of a geometric sequence?

First, use the formula aₙ = a₁ × rⁿ⁻¹. Substitute the first and fourth terms, solve for r cubed, and then take the cube root of the result to find r.

What is the common ratio if the first term of a geometric sequence is 5 and the fourth term is 320?

The common ratio is 4. This is calculated by dividing the fourth term (320) by the first term (5) and then taking the cube root of the result.

What is the process for solving a geometric sequence problem?

The process involves writing down the geometric sequence formula, identifying the given terms, substituting them into the formula, solving for the unknowns, and simplifying the expression to find the common ratio or another term.

What is the common ratio if the first term of a geometric sequence is 2 and the fourth term is 432?

The common ratio is 6. This is calculated by dividing the fourth term (432) by the first term (2), then taking the cube root of 216.

What is the next challenge question provided at the end of the video?

The challenge question asks: 'The first term of a geometric sequence is 1, and its common ratio is 3. Which term is 243?'

What mathematical property is used when dividing both sides of an equation to isolate the common ratio?

The multiplication property of equality is used when dividing both sides of an equation to isolate the common ratio, ensuring the equation remains balanced.

What is the purpose of the video lesson?

The purpose of the video is to teach how to solve geometric sequence problems by finding the common ratio and using the geometric sequence formula effectively.

Outlines

00:00

📚 Introduction to Solving Geometric Sequences

The paragraph introduces a video tutorial on solving geometric sequences, specifically focusing on finding the common ratio. The host encourages viewers to watch previous parts if they haven't, and then poses a challenge question: finding the common ratio of a sequence with a first term of 7 and a fifth term of 112. The formula for the nth term of a geometric sequence, a_n = a_1 * r^(n-1), is introduced. The host demonstrates how to apply this formula to find the common ratio by substituting the given terms and solving for r.

05:37

🔍 Example Walkthrough for Geometric Sequences

In this paragraph, the host provides a step-by-step solution to the challenge question from the introduction. By using the formula for the nth term of a geometric sequence, the host substitutes the given values and simplifies the equation to find the common ratio. The process involves dividing both sides of the equation by the first term and then solving for r^4. The host then explains how to find the fourth root of the resulting number to determine the common ratio.

11:16

🎓 Applying the Formula to Another Geometric Sequence

The host presents another example to demonstrate the process of finding the common ratio in a geometric sequence. This time, the sequence has a first term of 5 and a fourth term of 320. Using the same formula, the host substitutes the values and simplifies the equation to solve for r^3. After dividing both sides by the first term, the host finds r^3 and then takes the cube root to find the common ratio, which turns out to be 4.

16:18

🏁 Final Problem and Conclusion

The host concludes the tutorial by posing a final problem for the viewers to solve on their own: finding the common ratio of a sequence with a first term of 2 and a fourth term of 432. The host encourages viewers to pause the video and attempt the problem before revealing the solution. After a pause, the host provides the solution, demonstrating the use of the geometric sequence formula and the trial and error method to find that the common ratio is 6. The host thanks the viewers for watching and invites them to share the video and subscribe for more educational content.

🎉 Challenge Question and Farewell

In the final paragraph, the host presents an additional challenge question involving a geometric sequence with a first term of 1 and a common ratio of 3, asking viewers to determine which term equals 243. The host encourages viewers to try solving this on their own and looks forward to discussing it in the next video. The host thanks the viewers for watching, encourages them to share the lesson with friends, and reminds them to subscribe and turn on notifications for future videos. The video ends with a motivational message and a hashtag, '#YouMorePH', emphasizing the potential for learning and growth.

Keywords

💡Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the video, the instructor focuses on solving problems related to finding the common ratio in geometric sequences, such as with the first term of 7 and the fifth term of 112.

💡Common Ratio

The common ratio is the constant factor between consecutive terms in a geometric sequence. It is central to the video’s theme, where the speaker teaches how to find the common ratio given certain terms of the sequence, such as 7 and 112, using a specific formula.

💡First Term

The first term of a geometric sequence is the starting number in the series. In the video, the instructor uses the first term to set up problems for finding the common ratio. For instance, in one example, the first term is given as 7 and the process starts from there.

💡Fifth Term

The fifth term is the term in the geometric sequence that appears in the fifth position. The instructor uses this as part of a challenge problem to find the common ratio, such as in the example where the fifth term is 112.

💡Formula

In this context, the formula refers to the equation used to find a specific term in a geometric sequence: A_n = A_1 * r^(n-1). This formula is central to solving the problems in the video, where the instructor guides students on how to substitute given values into it to find unknowns like the common ratio.

💡A_n

A_n represents the nth term in a geometric sequence, calculated using the formula A_n = A_1 * r^(n-1). The video demonstrates how to use this variable to solve problems where specific terms, such as the fourth or fifth term, are given.

💡Exponent

The exponent in the geometric sequence formula refers to the power to which the common ratio (r) is raised. It’s calculated as (n-1), where n is the position of the term in the sequence. The instructor explains how to manipulate the exponent in the process of solving for the common ratio.

💡Multiplication Property of Equality

The multiplication property of equality is a method used to solve equations by multiplying or dividing both sides of the equation by the same number. In the video, the instructor uses this property when dividing both sides by the first term to isolate the common ratio.

💡Trial and Error Method

The trial and error method involves testing different values to find the solution to a problem. In the video, the instructor uses this method to find the cube root of 216 when solving for the common ratio, testing possible values like 6.

💡Challenge Question

A challenge question is presented to students as an exercise to test their understanding of the lesson. The instructor uses these questions throughout the video, such as asking viewers to find the common ratio of a sequence with a first term of 2 and a fourth term of 432.

Highlights

Introduction to a geometry sequence tutorial, part three.

Explanation of a challenge question involving a geometric sequence with a first term of seven and a fifth term of 112.

Presentation of the formula for the nth term of a geometric sequence: a_n = a_1 * r^(n-1).

Step-by-step solution to find the common ratio of the sequence with the given terms.

Division of both sides of the equation to isolate r, resulting in 16.

Clarification that the problem seeks r, not r^4, leading to the process of simplifying r^4 to r.

Final determination of the common ratio r as 2, by calculating the fourth root of 16.

Introduction to another example with a first term of five and a fourth term of 320.

Application of the geometric sequence formula to the new example.

Solution process involving division and simplification to find r as 4.

A pause for viewers to solve a similar problem with a first term of two and a fourth term of 432.

Solution to the viewer's problem, finding the common ratio r to be 6.

A challenge question is presented for viewers to solve on their own: finding the term number of a geometric sequence with a first term of one and a common ratio of three, where the term is 243.

Encouragement for viewers to share the tutorial with friends and to subscribe for more educational content.

The hashtag 'you more PH' is introduced as a call to action for continuous learning and growth.

Closing remarks and a prompt for viewers to join the next video in the series.

Transcripts

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my

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hi everyone welcome to las vegas

geometry sequence and this is now part

number three if you were not able to

watch part one in part two battery watch

it first

parameter today

i am so support number two nagentayo

is a challenge question i thought what

is the common ratio of the geometric

sequence

with first term seven and fifth term of

100

12 and i'll show you now how you're

going to solve

kapag uncommon and let's get

started ayan

all right so what is the common ratio of

the geometric sequence so

obviously common ratio young aryan

pinapahala

with the first term a sub 1 is 7 and

fifth

term of 112. so same process style

formula

given and solution so formula don't

forget champagne geometric

formula and geometric sequence

which is a sub n is equal to a sub 1

multiplied by r raised to n minus

1. okay so let's start with the given

a sub 1 and a sub one nothing i

first term so that is seven problem

next or nothing what is the common ratio

in champion

let's go to our solution so it's a

solution don't forget

right first the formula and after

writing the formula

or substitute these given so in a sub 1

making

seven be careful done one twelve da

pattern a sub five at the end a sub n

tap n is five or nothing as islam

unknown

and after that so at the moment you

exponent

that's five minus one

that's four i am and after

since atom or ah pinata

so divide by seven this process is the

multiplication property of equality

we're in

we're going to divide this by seven i

am so divide by seven both sides of the

equation

seven because this is one or raised to

four or just

raised to four that was dividing one

twelve and seven

will have today's result 16.

but the problem here is that angina

are right we're looking for r and not

r raised to four and how are we going to

get rid of this

um

a16

2 times two times two times two three

times three times three times three

hangars

so in this case ation ayan

so luma bastion 16 donsa

2 times 2 times 2 times 2 and we'll have

positive 16. so if it's a b

and four

and this is now our common ratio

and so that's how you deal with the

problem

okay let's try another example

what is the common ratio of the

geometric sequence with first term of

five

and fourth term of 320 again

same process formula given and solution

a common ratio given amongst a sub one

which is five because the first term

and n a fourth four

fourth term asap and nothing which is

320.

and so formula don't forget geometric

a sub n is equal to a sub 1 multiplied

by r raised to n

minus 1 and get the given a sub 1

unknown

5 first term r and r

r

that is 320

i am so we're done with formula and

given let's move now the solution

saying solution don't forget right first

the formula

a sub n is equal to a sub 1 multiplied

by r raised to n minus

one then after substituting

make sure in a c a sub one mugging five

make sure it's a magnitude four

and make sure it's a sub four a sub and

again 320 and by doing that

so four minus one is three magnitude

five or cube then after

since angola young are so this time

hindi paste

and this is mpe or multiplication

property of equality

we're in we're dividing this by five to

both sides of the equation

one r cube or just r cubed and

320 divided by five that is 64.

64 is equal to r cube

now same taiyanang process kanina

guessing our race to four

this time since

you brought the on again take note

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result and it is

64i i am

so 4 is equal to r or r is equal to

4 and that will now be the final answer

so as simple as that

now if you are dirty let's go to our

humor jail but if you are not

please go back to the discussion and

proceed to you more nothing foreign

all right so i'll give you one problem

here what is the common range of the

geometric sequence

with the first term of two and the

fourth term of 432

so i'll pause for a while now and i'll

get back to you to give the answer

but if you still need time feel free to

pause the video

your timer starts now

[Music]

all right times up so let us now answer

this

problem i answer what is the common

ratio of the geometric sequence with

first term of two and the fourth term of

four hundred thirty two

subpoena let's see that's r given

saturn's the first term

a sub one which is two and four is n

tapasan 432 is our a sub

4 so same process

formula given solution formula don't

forget about

these formula a sub n is equal to a sub

1 multiplied by r raised to n minus 1

and the given a sub one and a young a

sub one

first term that's two then

champion so question mark

that's fourth term number for that then

an a sub 4 that is 432

ayan then let's go to solution

so solution don't forget the formula

write that always

then substitute nothing yuma given after

that song a sub n magician 432

and a sub 1 2

4 or as east case

so 432 is equal to 2 multiplied by r

raised to 4 minus 1 so

r cube 432 is equal to

two r cube now

paramobile

divide nothing both sides by two i am

paramount

on two then this will become our cube

nalang one r cube or just r cube

then divide nothing and we'll have two

hundred

sixteen is equal to r cube now

oled panel

do the trial and error method but i'll

show you now for a new number in ion

that is uh sorry

six times six times six which is two

hundred

sixteen i am so after you making

is equal to r or r is equal to six and

that will now be

the final answer thank you so much for

watching guys i hope you learned

something today

but you will not end there i'll give you

another challenge question

and this will be about this one and the

first term of geometric sequence is one

and its common ratio is three which

term is two hundred forty three some

pina pahana

so try this on your own and let's check

out

so let's see each other on our next

video bye-bye

thank you thank you so much for watching

guys i hope you enjoyed and learned

something today

and if you did please do share this to

your friends who needs

this lesson the most as well and what

montgomery

subscribe and click on notification bell

updates videos nothing to see more ph

and don't forget that you deserve more

you can learn more

and you can be more hashtag

you more ph bye bye see you in our next

video

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you