GEOMETRIC SEQUENCE || GRADE 10 MATHEMATICS Q1
Summary
TLDRThis video introduces the concept of geometric sequences, exploring how a constant number is multiplied to obtain each subsequent term. It defines geometric sequences, identifies the common ratio, and demonstrates how to find the nth term. Through examples, the video explains how to determine whether a sequence is geometric and how to solve related problems. It also applies the concept to real-life scenarios, such as calculating the spread of infections in an outbreak. The video concludes with practice exercises to reinforce understanding.
Takeaways
- 🔢 A geometric sequence involves multiplying each term by a fixed number to get the next term, unlike arithmetic sequences where a number is added.
- 📈 The common ratio (r) in a geometric sequence is found by dividing any term by its preceding term.
- ✖️ To find the next term in a geometric sequence, multiply the last term by the common ratio.
- 🧮 The formula to find the nth term of a geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.
- 🔍 In the given examples, the sequences were analyzed to determine whether they were geometric by checking if the common ratio was consistent.
- ➗ A geometric sequence has a consistent common ratio across all terms, making it possible to predict future terms.
- ✅ The script provides a step-by-step guide to calculating the common ratio and identifying geometric sequences through various examples.
- 📊 The formula for the nth term is applied to real-life scenarios, such as calculating the spread of infections in a population.
- 🎯 The final example shows the application of the geometric sequence formula to predict the number of infections on the sixth day.
- 🔔 The video concludes by encouraging viewers to subscribe for more math tutorials.
Q & A
What is a geometric sequence?
-A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
How do you determine if a sequence is geometric?
-A sequence is geometric if there exists a common ratio 'r' that can be determined by dividing any term in the sequence by the term that precedes it.
What is the common ratio in the sequence 2, 8, 32, 128?
-The common ratio in the sequence 2, 8, 32, 128 is 4, as each term is obtained by multiplying the previous term by 4.
How do you find the next term in a geometric sequence?
-To find the next term in a geometric sequence, multiply the last known term by the common ratio.
What is the formula for finding the nth term of a geometric sequence?
-The nth term of a geometric sequence is given by the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.
In the sequence 5, 20, 80, what is the common ratio?
-The common ratio for the sequence 5, 20, 80 is 4, as each term is 4 times the previous term.
Is the sequence √2, 5√2, 3√2 a geometric sequence?
-No, the sequence √2, 5√2, 3√2 is not a geometric sequence because the ratio between consecutive terms is not constant.
What is the tenth term of the geometric sequence with first term 8 and common ratio 1/2?
-The tenth term of the geometric sequence with a first term of 8 and a common ratio of 1/2 is 1/64, calculated using the formula a_n = a_1 * r^(n-1).
How many people will be infected with measles on the sixth day if the number of infections grows geometrically with a common ratio of 2, starting with 4 infections on the first day?
-On the sixth day, there will be 128 people infected with measles if the number of infections grows geometrically with a common ratio of 2, starting with 4 infections on the first day.
How can you find the missing term in a geometric sequence if you know the first term and the common ratio?
-To find the missing term in a geometric sequence, multiply the first term by the common ratio raised to the power of the position of the missing term minus one.
Outlines
📚 Introduction to Geometric Sequences
In this section, the video introduces the concept of geometric sequences, where each term is obtained by multiplying the previous term by a constant number called the common ratio. The explanation includes how to identify the common ratio and calculate the next term in a sequence. Several examples are provided, such as finding the ratio between two terms and calculating subsequent terms in various sequences.
🧮 Examples and Identification of Geometric Sequences
This paragraph continues with more examples of sequences, demonstrating how to determine if a sequence is geometric by checking for a consistent common ratio. Examples include sequences with both positive and negative numbers, as well as fractions. The section also explains how to apply the formula for the nth term of a geometric sequence, using specific values to calculate terms like the tenth term in a sequence.
🚨 Application of Geometric Sequences in Real-World Scenarios
The final section discusses a real-world application of geometric sequences, using the example of the spread of measles during an outbreak. It explains how the number of infections can increase geometrically and walks through the calculation of the number of infections on the sixth day, using the geometric sequence formula. The video concludes by encouraging viewers to like, subscribe, and stay tuned for more educational content.
Mindmap
Keywords
💡Geometric Sequence
💡Common Ratio
💡Next Term
💡First Term (a₁)
💡Nth Term Formula
💡Sequence
💡Ratio
💡Geometric Series
💡Multiplication
💡Problem-Solving
Highlights
Introduction to geometric sequences and how they differ from arithmetic sequences by using multiplication instead of addition.
Definition of a geometric sequence and identification of the common ratio (denoted by 'r').
Explanation on how to find the common ratio by dividing any term in the sequence by the term that precedes it.
Example 1: Finding the common ratio and the next term in a geometric sequence with terms 1, 2, 4, and 8.
Example 2: Finding the common ratio and the next term in a geometric sequence with terms 80, 20, and 5.
Example 3: Finding the common ratio and the next term in a geometric sequence with terms 2, -8, 32, and -128.
Identifying if a sequence is geometric or not using four different examples.
Formula for finding the nth term of a geometric sequence: a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the number of terms.
Example problem: Finding the 10th term of a geometric sequence with first term 8 and common ratio 1/2.
Step-by-step calculation for finding the 10th term of the geometric sequence.
Another example: Finding the missing term in a sequence where the terms are 3, 12, 48, and the missing term needs to be calculated.
Second missing term example: Finding the missing term in a sequence where the terms are 32, 64, 128, and the first term needs to be calculated.
Application example: Using the geometric sequence formula to predict the number of infections during a measles outbreak.
Calculation of the number of infections on the 6th day of a measles outbreak, demonstrating real-world application of geometric sequences.
Conclusion and encouragement to like, subscribe, and follow for more math tutorials.
Transcripts
[Music]
in
previous lesson you already know
sequences
in which a certain number is added to
each term
to get the next term now you will
explore
sequences in which a certain number is
multiplied
so in this video we define geometric
sequence
and also we identify the common ratio in
the next term
of the sequence find the n term of the
geometric sequence
and solve problems involving geometric
sequence
we're going to understand how to use to
get the ratio to
the concept of ratio for this type of
sequence
so let's have a short activity find the
ratio of the second number to the first
number
for your example number one we have two
and
eight song enumeration in dalawa
individual second number this is the
first number so that will be
eight divide two so therefore young
raising ends is four
for number two we have negative three
and nine so divide like nothing young
nine k negative three so generation into
a negative three for number three we
have one
and one half so if you divide that
lattice one half k one so
the ratio is one half
a sequence is geometric if there exists
a number
r called the common ratio so we are the
uh we represent common ratio letter uh
the small letter
r the common ratio r can be determined
by dividing any term
in the sequence by the term that
precedes it okay let's have an example
identify the common ratio and the next
term
in the following sequences for number
one we have one
two four and eight of course
kanina is the first activity nathan
a diniscus
so therefore not young next term
emo multiplayer tends to eight
imo multiply not n c two k
eight parama young next term so the next
term is 16
since eight times two is equal to
sixteen
for number two we have eighty twenty
and five same process parama are not in
uncommon
term so 20 over 80 that is one-fourth
so parama young next term imma multiply
nothing
five one-fourth so that is
the next term is five over four since
five times one fourth so that is five
times one is
equal to five all over four
okay for number three we have two
negative 8
32 and negative 128 same
process to get the common ratio they
divide long nothing in second terms uh
first term so that is negative 8 over 2
that is equal to negative four so
therefore
your next term nothing immo multiply
nothing's
negative 128 k negative four and that is
five hundred twelve
okay you all know that the geometric
sequence
sequence the geometry my common racist
so i'll give you more examples to
identify
the given sequences if this sequence
is geometric or not
okay i have four examples here
so for the first example we have 5 20
80 twenty of course
but universe nothing in 20k pipe and the
answer is four
young 80 divided k20 the answer also is
four
okay 320 divided by 80. the answer is
also four
so
so therefore meron silang common ratio
so this
is a geometric sequence another
seven square root of two five square
root of two
three square root of two and square root
of two
so to check pagini write not tension
five square root of two by seven
square root of two omega one nothing is
five over seven because
nothing c three square root of two page
is not a geometric sequence
for number 3 we have 5 negative 10
20 and negative 40.
so same process divide nothing in second
terms of first term so
to check for my common ratio
negative 10 divide 5 that is negative 2
20 divided by negative
10 the answer is negative 2. at your
negative 40 divided 20
the answer also is negative two so
therefore marrow is uncommon ratio
so in number three not ten is a
geometric
sequence for number four we have ten
over three
ten over six ten over nine and ten over
fifteen
so checking muna natan kumai common
ratio given sequence
so apparently nothing in second terms of
first term
is one half ten over nine over ten over
six and so i got a two third
and ten over fifteen over ten over nine
and sagot i three fifth so mag
so therefore number four is not a
geometric sequence
the n term of a geometric sequence is
given by
a sub n so tandem for millennia
is equal to a sub 1 times r raised to n
minus 1.
take note r should not be equal to
zero so young a sub one nothing d though
that is our first term
and young are not then is young common
ratio net n
young n that is the number of terms
okay okay let's try to answer
this example what is the tenth term of
the geometric sequence
given eight four two
and one so identify muna natin yuma
given
so your common relation at n so four
over eight that is one half
your first term nothing that is eight so
we're using this formula so on gagawi ng
nathan is
okay so a sub 1 that is
8 times 1 half raised to
ten minus one back at ten so we have ten
terms
and not ten and then eight times one
half so ten minus one
is nine then one half raised to nine so
on gagavin
okay
so that is one half raised to nine so
that is one times
raised to negative one two raised to
nine is five hundred twelve
and then multiply that is eight over
five hundred
twelve and then eight over five hundred
twelve
is massive
so that is the final answer is one over
64.
okay so madali lang using the formula
so i have here another set of
uh activity or exercises
find the missing term in three twelve
forty eight
so on gaga in munich and of course
nathan and that is 48 times 4 that is
192. so i'm consumed at 192.
so you concentrate 192. same process
times k4 so the answer is 768.
another find the missing term in
okay blank flank 32 64 128.
and then 16 divide two that is eight so
in first term this is eight and
young 16.
okay i have here uh
one problem okay where in the geometric
sequence nah pedinating
apply during the initial pace of an
outbreak of
missiles the number of infection can
grow geometrically
if there were 4 8 16 on the first three
days
of an outbreak of the missiles how many
will be
infected on the sixth day okay gamete
formula
that is two and then your first term
nothing is four
and then young and nothing is six so
gamma in formula
substitute the value the first term is 4
and then this is the common ratio raised
to 6 minus 1
and then 4 times 2 raised to 5 and 2
raised to 5 is 32
32 times four that is 128 ebx a bn
there will be 128 people in
infected with measles on the sixth day
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the bell button
put updated ko for more video tutorial
this is your guide in learning your math
lesson your walmart channel
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