TAGALOG: Geometric Series #TeacherA #GurongPinoysaAmerika
Summary
TLDRIn this educational video, Teacher A introduces the concept of a geometric series, explaining the formula for calculating the sum of the first 'n' terms, S_n = a_1 * (r^(n-1)) / (r-1). The video demonstrates how to find the sum of the first eight terms of a sequence with a common ratio of 3, resulting in a sum of 3280. Teacher A also guides viewers through finding the sum of the first five terms of another sequence with a common ratio of 4, ending with a sum of 682. The tutorial is designed to be accessible and engaging, encouraging viewers to follow along and apply the concepts.
Takeaways
- 📚 The lesson focuses on geometric series, specifically finding the sum of the first n terms.
- ✏️ The formula for the sum of the first n terms of a geometric series is: S(n) = a₁ * (rⁿ - 1) / (r - 1), where a₁ is the first term and r is the common ratio.
- 📝 In the example sequence 1, 3, 9, 27, the first term (a₁) is 1, and the common ratio (r) is 3.
- 🔢 To find the sum of the first eight terms in this sequence, substitute a₁ = 1, r = 3, and n = 8 into the formula.
- 💡 The common ratio can be found by dividing consecutive terms in the sequence (e.g., 27 ÷ 9 = 3, 9 ÷ 3 = 3).
- 🧮 In the example, 3 raised to the 8th power is 6,561, and using the formula yields a sum of 3,280 for the first eight terms.
- ➗ The second example uses the sequence 2, 8, 32, where the first term is 2 and the common ratio is 4.
- 🔍 To find the sum of the first five terms of this sequence, substitute a₁ = 2, r = 4, and n = 5 into the formula.
- 🧠 For the second sequence, 4 raised to the 5th power is 1,024, and using the formula yields a sum of 682 for the first five terms.
- 📢 The teacher invites viewers to subscribe to their YouTube channel and follow their social media for more tutorials and updates.
Q & A
What is the formula for finding the sum of the first n terms in a geometric series?
-The formula for the sum of the first n terms (Sₙ) in a geometric series is Sₙ = a₁ * (rⁿ - 1) / (r - 1), where a₁ is the first term, r is the common ratio, and n is the number of terms.
In the given sequence (1, 3, 9, 27), what is the first term (a₁)?
-The first term (a₁) in the sequence is 1.
How is the common ratio (r) calculated in a geometric series?
-The common ratio (r) is calculated by dividing one term by the previous term. For example, in the sequence (1, 3, 9, 27), 3/1 = 3, 9/3 = 3, and 27/9 = 3. So, the common ratio is 3.
What are the steps to find the sum of the first eight terms in the sequence (1, 3, 9, 27)?
-1. Identify the first term (a₁ = 1) and the common ratio (r = 3). 2. Apply the formula Sₙ = a₁ * (rⁿ - 1) / (r - 1). 3. Plug in the values: S₈ = 1 * (3⁸ - 1) / (3 - 1). 4. Calculate 3⁸ = 6561, then S₈ = 1 * (6561 - 1) / 2 = 3280.
What is the sum of the first eight terms in the sequence (1, 3, 9, 27)?
-The sum of the first eight terms is 3280.
How is the common ratio determined in the sequence (2, 8, 32)?
-The common ratio (r) is determined by dividing successive terms. In the sequence, 8/2 = 4 and 32/8 = 4, so the common ratio is 4.
What is the sum of the first five terms in the sequence (2, 8, 32)?
-The sum of the first five terms is calculated using the formula S₅ = a₁ * (r⁵ - 1) / (r - 1). Plugging in the values: S₅ = 2 * (4⁵ - 1) / (4 - 1) = 2 * (1024 - 1) / 3 = 682.
What is the value of 4 raised to the power of 5 in the second example?
-4⁵ = 1024.
What is the significance of subtracting 1 in the geometric series formula?
-Subtracting 1 from rⁿ accounts for the fact that the formula finds the sum of terms from the first to the nth term, excluding higher terms beyond n.
What is the common ratio in the second example, and how does it affect the sum of the terms?
-The common ratio in the second example is 4. A larger common ratio leads to exponentially larger terms, which significantly increases the sum as the number of terms increases.
Outlines
👨🏫 Introduction to Geometric Series
In this introduction, the teacher welcomes viewers and introduces the topic of geometric series. The focus is on finding the sum of a geometric series using a specific formula: S(n) = a₁ × (r^n - 1) / (r - 1), where a₁ is the first term, r is the common ratio, and n is the number of terms. The example sequence used is 1, 3, 9, 27, and the goal is to find the sum of the first eight terms.
🔢 Step-by-Step Problem Solving
The video demonstrates how to solve a problem using the geometric series formula. The first step is to identify the given values: a₁ = 1, r = 3, and n = 8. The teacher explains how to calculate r by dividing consecutive terms in the sequence. The formula is then applied to calculate the sum: S(8) = 1 × (3^8 - 1) / (3 - 1), which results in 3280 as the sum of the first eight terms.
📊 Solving Another Example with Five Terms
In the second example, the teacher solves for the sum of the first five terms of a new geometric sequence: 2, 8, 32, and so on. The first term is a₁ = 2, and the common ratio is r = 4. Using the formula S(5) = 2 × (4^5 - 1) / (4 - 1), the teacher calculates the sum to be 682 for the first five terms. The explanation emphasizes following the steps methodically and applying the formula correctly.
📢 Invitation to Subscribe and Follow
The teacher concludes the video by inviting viewers to subscribe to the YouTube channel 'Teacher A Guru Financial America' and to follow the Facebook page of the same name. The channel offers educational content, including tutorial videos and activities related to financial topics. The teacher encourages viewers to stay updated with the latest videos and resources.
Mindmap
Keywords
💡Geometric Series
💡Common Ratio
💡Sum of the First n Terms (S_n)
💡First Term (a₁)
💡Exponentiation
💡Formula Substitution
💡Division
💡Multiplication
💡Sequence
💡Exponent
Highlights
Introduction to the concept of a geometric series and its formula.
Explanation of the formula for the sum of the first n terms of a geometric series.
Step-by-step guide to finding the sum of the first eight terms of a given sequence.
Identification of the first term (a sub 1) and common ratio (r) in the sequence.
Calculation of the common ratio by dividing consecutive terms.
Application of the geometric series formula to find the sum of the first eight terms.
Explanation of the calculation process for the sum of the first eight terms.
Final result of the sum of the first eight terms in the sequence.
Introduction to the second example involving a different geometric sequence.
Step-by-step guide to finding the sum of the first five terms of the new sequence.
Identification of the first term and common ratio for the second sequence.
Application of the geometric series formula to find the sum of the first five terms.
Explanation of the calculation process for the sum of the first five terms.
Final result of the sum of the first five terms in the second sequence.
Emphasis on following the step-by-step procedure for solving geometric series problems.
Invitation to subscribe to the YTC channel for more educational content.
Encouragement to like the Facebook page for updates on latest videos and activities.
Conclusion of the tutorial with a reminder to join for the next video.
Transcripts
good morning guys teacher a here and
welcome to golden eyes america so for
today unless you're not in eye geometric
series we're in capacity geometric
series
terms
sequence
so to find
some
formula
so s of n is equal to a sub 1 times
r raised to n minus one over r minus one
we're in c s sub n union
terms
c a sub one first term
c r uncommon ratio at c
so we have here find the sum of the
first eight terms in this given sequence
we have one three
nine twenty seven
so step one okay identify nothing you're
given
nothing
formula
and terms
eight terms so it's the end
next c a sub one
x b hen you first terminated starting
sequence which is one
and then cr
in common ratio
and the bio number multiplication three
three online multiplicity three is a
number nine nine bucks multiplicity
number nine twenty seven
shortcut a divide so we have 27 divided
by nine
that will give us three
nine divided by three again that's three
three divided by one that's three
so therefore i'm adding r in common
ratio nothing is three okay
so that is our first step
second step is nothing in formula
so s of n
is equal to
a sub 1 times
r raised to n and then minus 1
over
r minus 1.
and then after that it plug in number of
values so s sub under the end eight
which means sum num eight in the terms
equals c a sub one is one
times
3
raised to n and n not in i 8
and then minus 1
over
c r naught and let i 3
[Music]
so 1 times
r to the 8th episode 3 ir sorry 3 to the
eight means three times three times
three now eight times so one of these is
nothing you will multiply c three
so three to the eighth is equal to
six thousand five hundred
sixty-one
and then copyright minus one
over c three minus one eight two
so solving this we have one
times
six thousand five hundred sixty-one
minus one i six thousand five hundred
sixty
divided by two divide not that
uh six thousand five hundred sixty
divided by two is three thousand
two hundred eighty
multiplied by 1 is still 3280.
so therefore a sum
num first
eight terms now given a sequence i 3280.
now
a sub 1
actually
so
a sub 1 times parenthesis r
to the n minus one and then all over
r minus one so panda bayon you're not
going to example number two
okay number two find the sum of the
first five terms given the sequence 2 8
32 and so on and so forth using the
formula s sub n is equal to a sub 1
times r raised to the n minus 1 all over
r minus one so first step
nothing given
in my important
quantities
i
basically starting problem five
experience five terms
next you're adding a sub one your first
term so given i
two
and then i'm adding r which is common
ratio divide nothing
8 divided by 2
so 8 divided by 2 is 4
32 divided by 8 is also 4. so raise your
nut and i four
yan that's our first step
second step is formula
s sub n is equal to a sub 1 times
r raised to n
minus one
all over
r minus one
so after writing down the formula it
plug in or substitute nothing more
values so s sub n we have
s sub
five
so meaning sum not five terms
is equal to a sub one not in a two
parenthesis i'm are not in a four
raised to the n which is five
and then minus one
over
c r uh and i four
minus one
then solve nothing
first evaluating my exponent so we have
two
four to the fifth means four times four
times four times four i
1024
and then minus one
over
four minus one i three
so nothing in a cell
one thousand
twenty-four
minus one so hopefully i'm going to see
two i
1023
then divided by 3
so 2 times 1023i
2046
divided by three therefore and sum now
first five terms now given a sequence i
682
yan
so
okay so just follow the step-by-step
procedure
and please i am inviting you number
subscribes
ytc teacher a girl fitness america and
also please like my fp page same name
teacher a guru financial america updated
my latest videos
pictures the modules are activities
woman tutorial video okay that's it for
today see you in my next video
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