Geometric Series and Geometric Sequences - Basic Introduction
Summary
TLDRThis educational video script explores the concepts of geometric sequences and series, distinguishing them from arithmetic ones by their common ratio versus common difference. It explains how to calculate the nth term and the partial sum of a geometric series, highlighting the formulae involved. The script also discusses the arithmetic and geometric means, provides examples of writing equations between terms, and covers the sum of infinite geometric series, emphasizing the convergence criteria. Practice problems are included to reinforce the concepts.
Takeaways
- 📚 The difference between a geometric sequence and a series is that a geometric sequence is a list of numbers with a common ratio between terms, while a geometric series is the sum of the numbers in a geometric sequence.
- 🔢 In a geometric sequence, each term is found by multiplying the previous term by a common ratio, whereas in an arithmetic sequence, each term is found by adding a common difference to the previous term.
- 🧮 The formula to find the nth term of a geometric sequence or series is \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
- 📈 The partial sum formula for a geometric series is \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \), which calculates the sum of the first \( n \) terms.
- ⚠️ The sum of an infinite geometric series can only be calculated if the series converges, which occurs when the absolute value of the common ratio \( r \) is less than 1.
- 🔁 The sum of an infinite geometric series that converges is given by \( S_{\infty} = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
- 🔢 The arithmetic mean (Ma) of two numbers in an arithmetic sequence is the average, while the geometric mean (Mg) of two numbers is the square root of their product.
- 📉 To relate terms within a geometric sequence, multiply the previous term by the common ratio raised to the power of the term's position difference.
- 📝 When identifying a sequence or series, look for a common ratio (geometric) or common difference (arithmetic), and determine if it is finite or infinite based on whether it has an end or continues indefinitely.
- 📊 Practice problems in the script illustrate how to calculate terms of a geometric sequence, write general formulas, and determine the type of sequence or series based on given patterns.
- 🤓 Understanding the properties of geometric sequences and series is essential for solving problems involving series convergence, term calculation, and sum determination.
Q & A
What is the main focus of the video?
-The video focuses on geometric sequences and series, explaining the difference between them, how to identify them, and how to calculate various terms and sums within these sequences and series.
What distinguishes a geometric sequence from an arithmetic sequence?
-A geometric sequence is distinguished by a common ratio between consecutive terms, whereas an arithmetic sequence has a common difference between terms. In a geometric sequence, each term is found by multiplying the previous term by the common ratio.
How is the common ratio of a geometric sequence calculated?
-The common ratio is calculated by dividing any term in the sequence by the previous term. For example, if the sequence is 3, 6, 12, ..., the common ratio is 2, since 6 divided by 3 equals 2.
What is the formula to find the nth term of a geometric sequence or series?
-The formula to find the nth term of a geometric sequence or series is given by \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
How do you calculate the sum of the first n terms of a geometric series?
-The sum of the first n terms of a geometric series is calculated using the formula \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \), where \( S_n \) is the sum, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
What is an infinite geometric series and when does it converge?
-An infinite geometric series is a series that continues indefinitely. It converges when the absolute value of the common ratio \( r \) is less than 1, meaning the terms get smaller and smaller, allowing the series to sum to a finite value.
How is the sum of an infinite geometric series calculated?
-The sum of an infinite geometric series is calculated using the formula \( S_{\infty} = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term and \( r \) is the common ratio, provided that the absolute value of \( r \) is less than 1.
What is the difference between the arithmetic mean and the geometric mean of two numbers?
-The arithmetic mean of two numbers is the average, found by adding the numbers and dividing by 2. The geometric mean is the square root of the product of the two numbers, reflecting the middle term in a geometric sequence.
How can you determine if a sequence is arithmetic or geometric by looking at its terms?
-A sequence is arithmetic if there is a common difference between consecutive terms, while it is geometric if there is a common ratio by which you multiply one term to get the next.
What is the relationship between the terms in a geometric sequence defined by a recursive formula?
-In a geometric sequence defined by a recursive formula, each term is found by multiplying the previous term by the common ratio, represented as \( a_n = r \times a_{n-1} \).
Can you provide an example of how to write the first five terms of a geometric sequence given the first term and the common ratio?
-Certainly. If the first term \( a_1 \) is 2 and the common ratio \( r \) is 3, the first five terms would be calculated as follows: \( a_1 = 2 \), \( a_2 = 2 \times 3 = 6 \), \( a_3 = 6 \times 3 = 18 \), \( a_4 = 18 \times 3 = 54 \), and \( a_5 = 54 \times 3 = 162 \).
Outlines
📚 Introduction to Geometric Sequences and Series
This paragraph introduces the topic of geometric sequences and series, distinguishing them from arithmetic sequences by the presence of a common ratio in geometric sequences versus a common difference in arithmetic sequences. It provides an example sequence and explains how to identify the common ratio. The paragraph also explains the difference between a sequence and a series and introduces the formula for finding the nth term of a geometric sequence or series, which is the first term multiplied by the common ratio raised to the power of (n-1). An example calculation is provided to illustrate the formula's application.
🔍 Understanding Geometric Series and Means
The second paragraph delves into geometric series, which are the sums of numbers in a geometric sequence, and provides the formula for calculating the partial sum of a geometric series. It contrasts finite and infinite geometric series, explaining that infinite series continue indefinitely. The paragraph also introduces the concepts of arithmetic mean (average of two numbers) and geometric mean (square root of the product of two numbers), using examples from arithmetic and geometric sequences to illustrate how to find these means.
🔢 Writing Equations and Summing Infinite Geometric Series
This paragraph discusses how to write equations between terms within a geometric sequence, emphasizing the role of the common ratio in relating terms. It explains that to move from one term to another, you multiply by the common ratio raised to the power of the difference in their positions. The paragraph also addresses how to calculate the sum of an infinite geometric series, noting that it converges and can be summed if the absolute value of the common ratio is less than one. Examples are provided to demonstrate these concepts.
📝 Practice Problems on Geometric Sequences
The fourth paragraph presents practice problems involving writing the first five terms of given geometric sequences, using the common ratio to find subsequent terms. It also includes writing a general formula for the nth term of a geometric sequence and calculating the value of a specific term, such as the eighth term. The paragraph guides through the process of identifying the common ratio and applying it to find terms in the sequence.
📉 Describing Patterns in Sequences and Series
In this paragraph, the task is to describe patterns in numbers as either arithmetic or geometric, finite or infinite, sequences or series. The explanation involves identifying whether there is a common difference or ratio and the presence of plus signs or commas to determine if it's a series or sequence. It also discusses whether the pattern continues indefinitely or has a clear end to classify it as finite or infinite.
🧮 Calculating Sums of Geometric Sequences
The sixth paragraph focuses on calculating the sum of the first ten terms of a geometric sequence and the sum of an infinite geometric series. It explains the formula for the sum of the first n terms of a geometric sequence and how to apply it, as well as the criteria for an infinite geometric series to converge (absolute value of the common ratio less than one). The paragraph provides step-by-step calculations for both scenarios.
🏁 Conclusion on Summing Infinite Geometric Series
The final paragraph wraps up the discussion on infinite geometric series, providing a formula for calculating their sum and emphasizing the condition for convergence (absolute value of the common ratio must be less than one). It includes a calculation example to find the sum of a specific infinite geometric series, demonstrating the process clearly.
Mindmap
Keywords
💡Geometric Sequence
💡Geometric Series
💡Common Ratio
💡Arithmetic Sequence
💡Partial Sum
💡Infinite Geometric Series
💡Arithmetic Mean
💡Geometric Mean
💡Recursive Formula
💡Convergence
💡Divergence
Highlights
Introduction to the difference between geometric sequences and series.
Example of a geometric sequence with a common ratio.
Explanation of how to distinguish between arithmetic and geometric sequences.
Formula for calculating the nth term of a geometric sequence or series.
Demonstration of calculating the fifth term of a geometric sequence.
Partial sum formula for the sum of the first n terms of a geometric series.
Calculation of the sum of the first five terms of a geometric series.
Identification of infinite geometric series and their properties.
Concept of arithmetic and geometric means and their differences.
Finding the arithmetic mean between terms in an arithmetic sequence.
Calculating the geometric mean between terms in a geometric sequence.
Writing equations between terms within a geometric sequence using common ratios.
Formula for calculating the sum of an infinite geometric series and its conditions for convergence.
Examples of calculating the sum of infinite geometric series with different common ratios.
Practice problems for writing the first five terms of given geometric sequences.
Writing a general formula for the nth term of a geometric sequence and calculating specific terms.
Describing patterns of numbers as arithmetic or geometric, finite or infinite, sequence or series.
Finding the sum of the first ten terms of a geometric sequence with a given common ratio.
Summation of an infinite geometric series with a specific common ratio and its convergence criteria.
Transcripts
in this video we're going to focus on
geometric sequences and series
so first let's discuss the difference
between
a geometric sequence
and
a geometric series
what do you think the difference is
here's an example of a geometric
sequence
the numbers 3
6
12
24
48 and so forth
a geometric sequence
is different from
an arithmetic sequence
such as
this one here
in that a geometric sequence has a
common ratio versus a common difference
if you take the second term and divide
it by the first term
six divided by three is two
you're going to get the common ratio
if you take the third term divided by
the second term you'll get the same
common ratio 12 divided by 6 is 2.
so that's the defining mark of a
geometric sequence
in an arithmetic sequence there's a
common difference
if you take the second term and subtract
it from the first term
eight minus five is three
if you take the third term subtracted by
the second
eleven minus eight is three
so that's how you could distinguish an
arithmetic sequence from a geometric
sequence an arithmetic sequence has a
common difference between terms
a geometric sequence has a common ratio
between terms
within an arithmetic sequence you're
dealing with addition and subtraction
for a geometric sequence you're dealing
with multiplication and division between
terms
so now that we know what a geometric
sequence is and how to distinguish it
from an arithmetic sequence what is the
geometric series
a geometric series is basically
the sum of the numbers in a geometric
sequence
so 3 plus 6 plus 12
plus 24
and so forth
would be a geometric series
this is the first term this is the
second term this is the third
and so forth
now the formula you need to calculate
the f term of a geometric sequence
or series
it's
the first term a sub 1
times the common ratio r raised to the n
minus 1. so for instance
let's just make a note that r is equal
to 2.
let's say we want to find the value of
the fifth term we know the fifth term is
48 but let's go ahead and calculate it
so you can see how this formula works
so the first term is three
r
the common ratio is two
and n is the subscript here we're
looking for the fifth term so
n is five
so five minus one is four
two to the fourth power if you multiply
two four times two times two times two
times two that's 16.
16 times 3 is 48
so that's the function of this formula
it gives you the value of the f term so
you can find the value of the eighth
term the 20th term and so forth
the next equation you need to be
familiar with
first let's get rid of this
the next equation is the partial sum
formula
the partial sum of a geometric series
is the first term times 1 minus r raised
to the n
over
1 minus r
so let's say that we want to find the
sum of the first five terms
this is going to be 3
plus 6
plus 12
plus 24.
plus 48
go ahead and plug that into a calculator
so for the first five terms i got the
partial sum as being 93.
now let's confirm that with this
equation
so let's calculate s sub 5
the first term is 3
times 1 minus r r is 2 n is 5
divided by 1 minus r so that's 1 minus
2.
2 to the fifth power
that's 32
1 minus 2 is negative 1.
now 1 minus 32 is negative 31.
3 times negative 31 that's negative 93
but divided by negative 1 that becomes
positive 93
so we get the same answer
so anytime you need to find
the sum
of a finite series
you could use this formula
so
this series here is finite
we're looking for the sum of the first
five terms there's beginning and there's
an end
this series here
is
not finite it's an infinite geometric
series
the reason being is because of the dot
dot that we see here it goes on forever
it doesn't stop at the fifth term it
keeps on going to infinity so it's an
infinite geometric series
this
is an infinite
geometric sequence
it's a sequence that goes on from
forever and it's geometric so make sure
you can identify
if a sequence is arithmetic geometric is
it finite infinite is it a sequence or a
series
now the next thing we need to talk about
is
the arithmetic mean and the geometric
mean
let's call the arithmetic mean m a
the arithmetic mean is simply the
average of two numbers
the geometric mean let's call it mg
is the square root of the product of two
numbers
so let's go back to
the arithmetic sequence that we had here
if we wanted to find
the arithmetic mean between the first
and the third term it will give us the
middle number the second term
if you average 5 and 11 and divide by 2
using this formula
you're going to get 16 over 2
which is eight
so thus when you find the arithmetic
mean of the first term
and the third term
you're going to get the second term
because the average of one and three is
two
now
let's find the arithmetic mean between
the first and the fifth term
this will give us
the middle term 11.
so if we were to add up a1
and a5
and then divided by 2 if we were to get
the average
we would get a3 the average of 1 and 5
is three
so let's add five and seventeen and then
divide by two
five plus seventeen is twenty two
twenty two divided by two is eleven
so that's the concept of the arithmetic
mean whenever you take
the arithmetic mean of two numbers
within an arithmetic sequence you get
the middle term of that of those two
numbers that you selected
now the same is true for a geometric
sequence
if we were to find the geometric mean
between 3 and 12
we would get the middle number 6.
if we wanted to find the geometric mean
between 3 and 48 we would get the middle
number 12.
so let's confirm that
let's find the geometric mean
between a1 and a3
so the first term is 3
the third term is 12.
3 times 12
is 36
the square root of 36 is 6.
so we get the middle number
now let's find the geometric mean
between the first term and the fifth
term
so we should get 12 as an answer
so the average of one and five one plus
five is six divided by two is three so
we should get a sub three
the first term is 3 the last term or the
5th term is 48.
now what's 3 times 48
if you're not sure what you could do is
break it up into
smaller numbers
48 is three times sixteen
three times three is nine so you have
the square root of nine times the square
root of sixteen
the square root of nine is three the
square root of sixteen is four three
times four is twelve
so the geometric mean
of 3 and 48
is the middle number in the geometric
sequence which is 12.
now sometimes you need to be able to
write equations between
terms within a geometric sequence
for instance if you want to relate the
second equation
to the first equation you need to
multiply by r
i mean the second term to the first term
if you want to relate the fifth term to
the second term you need to multiply by
r cubed
to go from the second term to the fifth
term you need to multiply it by r three
times
if you multiply six by r you're going to
get 12. if you multiply 12 by r you get
24.
24 by r you get 48.
so to go from the second term to the
fifth term you need to multiply by r
cubed
and the reason why it's cube is because
the difference between five and two is
three
and you could check that so if you take
the second term which is six multiply it
by two to the third that's six times
eight which is 48 and that gives you the
fifth term
so if i want to relate the ninth term
to the fourth term
how many r values do i got to multiply
the fourth term to get to the ninth term
nine minus four is five
so i gotta multiply the fourth term by r
to the fifth power to get the ninth term
so make sure you know how to write those
formulas
so we've discussed calculating the sum
of a finite series
just review if you want to calculate the
sum
of a finite series one that has a
beginning and an end
you would use this formula
now what about the sum
of an infinite series
how can we find that
what's the formula
that we need to calculate s to infinity
it's basically this same formula but
without that part it's a one over one
minus r
so here's two examples
of an infinite geometric series this is
one of them
and this one is going to be another one
eight
four
two
one
one half
and so forth
we can't calculate the sum of both
infinite geometric series
for this one r is two
so r
or rather the absolute value of r is
greater than one
when that happens
the geometric series diverges
which means you can't calculate the sum
because it doesn't
it doesn't converge to a specific value
if you keep adding these numbers
it's not going to converge to a value
it's going to get bigger and bigger and
bigger
so the series diverges
if you try to calculate it let's say you
plugged in 1 for a1
and
2 for r it's not going to work
you get 3 over negative 1 which is
negative 3 and clearly that's not the
sum of this series
the fact that
you get a negative sum from positive
numbers tells you something is wrong
so this formula
doesn't work if the series diverges it
only works if the series converges and
that happens
when the absolute value of r is less
than one
if we focus on this particular infinite
geometric series
notice the value of r
if we take the second term divided by
the first term
four over eight is one half
if we take the third term divided by the
second term
two over four reduces to one half
so that's the value of r
so for that particular series we could
say that
the absolute value of r which is one
half
that's less than one
therefore
the series
converges
which means we can calculate a sum it
has a finite sum even though the numbers
get smaller and smaller and smaller
now let's calculate the sum
so the sum
of an infinite number of terms of this
geometric series is going to be the
first term a sub 1 which is 8
over 1 minus r
where r is a half
one minus one half is one half
so multiplying the top and bottom by two
we get 16 on top
these two will cancel we get one
so the sum of this infinite geometric
series that converges is 16.
so that's how you can calculate the sum
of an infinite geometric series
the series must converge and for that to
happen the absolute value of r has to be
less than one
if it's greater than one the series will
diverge and you won't be able to
calculate the sum
now let's work on some practice problems
write the first five terms of each
geometric sequence shown below
so let's start with the first one
the first term is two
to find the next term we need to
multiply
the first term by the common ratio the
second term is equal to the first term
times the common ratio
so 2 times 3 is 6.
and then to get the third term we just
got to multiply the second term by the
common ratio
6 times 3 is 18
18 times 3 is 54
and then 54 times 3 that's 162.
so that's the answer for
number one
let's move on to number two
the first term is 80.
the common ratio is one-half
so we're going to multiply 80 by a half
half of 80 is 40
half of 40 is 20
half of 20 is 10
half of 10 is five
so those are the first five terms for
the second geometric sequence
now let's move on to number three
so the first term is six
to find the next term we need to
multiply six by negative two
so this is going to be negative twelve
negative twelve times negative two is
positive 24
and then it's just going to alternate
so whenever you see a sequence a
geometric sequence with
alternating signs
then you know that the common ratio must
be negative
number two
write the first five terms of the
geometric sequence defined by the
recursive formula shown below
so we're given the first term
when n is 2 we have that the second term
is equal to negative 4
times the first term
and we know that the second term
is the first term times r
so therefore r
the common ratio must be negative 4.
so anytime you need to write a recursive
formula of a geometric sequence it's
going to be a sub n is equal to r
times
the previous term a sub n minus 1.
the next term is always the previous
term times the common ratio
so the common ratio
is this number negative four
so once we have the first term in the
common ratio we can easily write out the
sequence so the first term is negative
three
the second term will be negative three
times negative four
which is twelve
the third term
will be
net twelve times negative four which is
negative forty eight
the fourth term is negative 48 times
negative 4
which is 192
and then the fifth term
192 times negative 4
is negative
768
so that's how we can write the first
five terms of the geometric sequence
defined by recursive formula it's by
realizing that this number is the common
ratio
write a general formula that gives the f
term of each geometric sequence
and then calculate the value of the
eighth term
of each of those geometric sequences
so let's start with number one
so we have the number 6
24
96
384 and so forth
the first thing we need to do is
calculate the common ratio
so let's divide the second term by the
first term
dividing 24 by six we get four
now just to confirm that
this is indeed a geometric sequence
let's take the third term and divide it
by this the second term not the first
one
so 96 divided by 24
and that is also equal to 4.
so we have a geometric sequence here
in order to write the formula
all we need is the value of the first
term and the common
ratio so we could use this equation the
f term is going to be equal to the first
term times r
raised to the n minus one
the first term being six
r is four
so we can write it as a sub n is equal
to 6
times 4 raised to the n minus 1.
so this is the answer for part a for the
first
sequence
now let's move on to part b let's
calculate the value
of the eighth term
so we just got to plug in 8 into n
so it's 6
times 4 raised to the eight minus one
eight minus one is seven
four raised to the seventh power is
sixteen thousand three hundred eighty
four
times six
this gives us ninety eight thousand
three hundred and four
so that is the value of the eighth term
and you could confirm it
if you keep
multiplying these numbers by 4 you're
going to get it
384
times 4
that's
15 36 that's the fifth term
if you times it by 4 again
you get sixty one forty four
times four you get
twenty four five seven six
and then times four gives you this
number
now let's move on to number two
so we have the sequence
5
negative 15 45
negative 135 and so forth
so the first term is five
the common ratio
which can be calculated by taking the
second term divided by the first term
that's negative 15 divided by five
that's negative three
r is also equal to the third term
divided by the second term
so that's 45
over
negative 15 which is negative three
so the value of the first term is five
r
is negative three
so now let's go ahead and write a
general formula that gives us the nth
term
so a sub n is going to be a sub 1 times
r raised to the n minus 1.
the first term is 5
r is negative three
so this right here is the answer
for part a
now part b calculate the value of the
eighth term
so let's replace n with eight
negative three raised to the seventh
power
that's negative two thousand one hundred
and eighty-seven
multiplying that by five
this gives us negative ten thousand
nine hundred and thirty-five
so that's the final answer for part b
number four
describe each pattern of numbers as
arithmetic or geometric finite or
infinite sequence or series
let's look at the first one
do we have a common difference or common
ratio
going from 4 to 8
we increase by four
from eight to twelve that's an increase
by four
and twelve to sixteen
so we're constantly adding four we're
not multiplying by four
so therefore
we have a common difference and not a
common ratio
so because we have a common difference
this is arithmetic
not geometric
we're dealing with addition rather than
multiplication
now is this a sequence or series
we're not added numbers so
we have a sequence
if you see a comma between the numbers
it's going to be a sequence if you see a
plus you're dealing with a series
now
is this sequence finite or infinite
it has a beginning and it has an end
we don't have
dots that indicate that it goes on
forever so this is finite so we have a
finite arithmetic sequence
for number one now let's move on to
number two
going from 90 to 30 that's a difference
of negative 60. going from 30 to 10
that's a difference of negative 20.
so we don't have a common difference
here
if we divide the second term by the
first term
this reduces to one-third
if we divide the third term by the
second term that's also
one-third so what we have here
is a common ratio
rather than a common difference
so the pattern of numbers is geometric
not arithmetic
we're multiplying by 1 3
to get the second term from the first
term
now are we dealing with a sequence or
series
so we don't have a plus sign between a
number so we have a comma
so we're dealing with a sequence
and this sequence has no end it goes on
forever
so what we have here is an infinite
geometric sequence
now for number three
we could see that we have a common ratio
of two
five times two is ten ten times two is
twenty twenty times two is forty
so this is geometric
now there's a plus between the numbers
so this is going to be a series not a
sequence
and this sequence i mean the series
rather comes to an end the last number
is 80. so it's not infinite but it's
finite
so we have a finite geometric series
for the last one we can see that we have
a common difference of negative four
fifty minus four
is forty six forty six minus four is 42
so this is going to be arithmetic
we have a plus between a number so it's
a series
and it goes on forever
so it's infinite
so we have an infinite arithmetic series
number five
find the sum of the first ten terms of
the geometric sequence shown below
so the first term is 7
the common ratio
negative 14 divided by 7
that's negative 2.
28 divided by negative 14 is also
negative 2.
so now that we know the first term in
the common ratio we can calculate the
sum
using this formula
so it's the first term times one minus r
raised to the n
over one minus r
so the sum of the first 10 terms
it's going to be the first term 7
times 1 minus
negative 2
now it's raised to the n power n is 10
and then divided by 1
minus r
negative two
raised to the tenth power
that's positive one thousand
twenty four
and here we have one minus negative two
which is one plus two and that's three
one minus
ten twenty four
that's negative one thousand
twenty three
seven times negative ten twenty three
divided by three
that's negative two thousand three
hundred and eighty seven
so that's the final answer
try this problem find the sum of the
infinite geometric series
so we have the numbers 270
90
30
10
and so forth
so we can see that the first term a sub
1
that's 270.
the common ratio if we divide 90 by 70
what does that simplify to
i mean 90 by 270.
well we could cancel a 0 so we get 9
over 27
9 is 3 times 3 27 is 9 times three
so that becomes true over nine
three we can write that as three times
one nine is three times three this is
one third
so that's going to be the common ratio
and of course if you divide
30 by 90 you also get one-third
so now that we have the first term and
the common ratio
we can now calculate
the sum of the infinite geometric series
using this formula it's going to be a
sub 1 over 1 minus r
by the way
this particular infinite geometric
series does a converge converger diverge
the absolute value of r
is less than one it's one third which is
about 0.333 or 0.3 repeating
so because it's less than 1
the infinite geometric series
converges
we can calculate the sum the sum is
finite
so let's go ahead and calculate that sum
the first term is 270
r
is one third
so what's one minus one third
one if you multiply it by three over
three
you get three over three minus one over
three which is
two over three
now what i'm going to do is i'm going to
multiply the top and bottom by three
these threes will cancel
so it's going to be 270
times three divided by two
two seventy divided by two is one thirty
five one thirty five times three
and that's going to be four oh five
so that is the sum
of this infinite geometric series
you
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