Grade 10 Math Q1 Ep7: Finding the nTH term of a Geometric Sequence and Geometric Means
Summary
TLDRIn 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
📚 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.
🔍 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.
🧩 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.
🎓 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.
📘 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.
Mindmap
Keywords
💡Geometric Sequence
💡Common Ratio
💡nth Term
💡Geometric Mean
💡Logical Reasoning
💡Critical Thinking
💡Self-learning Module
💡Formula
💡Sequence
💡Terms
💡Ratio
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
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