[TAGALOG] Grade 10 Math Lesson: SOLVING GEOMETRIC SEQUENCE (Part III)- FINDING THE COMMON RATIO
Summary
TLDRIn 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
📚 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.
🔍 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.
🎓 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.
🏁 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.
Mindmap
Keywords
💡Geometric Sequence
💡Common Ratio
💡First Term
💡Fifth Term
💡Formula
💡A_n
💡Exponent
💡Multiplication Property of Equality
💡Trial and Error Method
💡Challenge Question
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
[Music]
[Music]
my
[Music]
[Music]
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
[Music]
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
[Music]
you
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