Unit 1 Lesson 2 Practice Problems IM® Algebra 2TM authored by Illustrative Mathematics®
Summary
TLDRThe video script is an educational tutorial on geometric sequences, growth factors, and patterns in sequences. It explains how to find subsequent terms in a geometric sequence by multiplying by a constant growth factor. It also demonstrates calculating growth factors by dividing successive terms and applies this concept to a credit card debt example with compounding monthly interest. The script further explores the Sierpinski triangle, illustrating the pattern of shaded triangles and the relationship between the number of triangles and their area over iterations. Additionally, it discusses graphing these patterns and creating custom sequences based on given rules, offering insights into mathematical sequences and their applications.
Takeaways
- 🔢 In a geometric sequence, each term is found by multiplying the previous term by a constant factor, known as the growth factor.
- 📈 The growth factor can be determined by dividing a term by its preceding term in the sequence.
- 💳 A credit card balance with a 2% monthly interest rate increases by 102% of the previous month's balance, calculated by multiplying the balance by 1.02 each month.
- 📊 The number of shaded triangles in a Sierpinski triangle increases exponentially with each step, while the area of each triangle decreases.
- 📉 The graphs of the number of shaded triangles and the area of each triangle in a Sierpinski triangle demonstrate different growth patterns; the number of triangles increases while the area per triangle decreases.
- 🔑 The rule 'four less than three times the previous number' is used to generate sequences where each term is calculated by multiplying the previous term by three and then subtracting four.
- 📐 The Sierpinski triangle is constructed by recursively removing the middle triangle from a set of four congruent triangles, creating a fractal pattern.
- 📘 The area of each triangle in the Sierpinski triangle sequence is calculated by dividing the total area by the number of triangles at each step.
- 💡 The script demonstrates how to apply geometric sequence concepts to real-world scenarios, such as calculating compound interest on a credit card.
- 📌 The script illustrates the process of creating and analyzing sequences using different rules, showcasing the versatility of geometric sequences.
Q & A
What is the definition of a geometric sequence?
-A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
How do you find the next three terms of a geometric sequence if the first two terms are given?
-To find the next three terms of a geometric sequence, you multiply each term by the common ratio to get the subsequent term. For example, if the first term is 1 and the second term is 4, and the common ratio is 2 (since 1*2=2 and 2*2=4), the next three terms would be 4*2=8, 8*2=16, and 16*2=32.
What is the growth factor of a geometric sequence and how do you calculate it?
-The growth factor, also known as the common ratio, is the number by which you multiply to get from one term in a geometric sequence to the next. It can be calculated by dividing any term by its preceding term.
How does the growth factor affect the terms in a geometric sequence?
-The growth factor determines the rate at which the terms in a geometric sequence increase or decrease. A growth factor greater than 1 will increase the terms, while a growth factor less than 1 will decrease them.
If a credit card balance of $1000 has a 2% monthly interest rate and no payments are made, how does the balance change over time?
-With a 2% monthly interest rate, the balance increases by 2% each month. This means the balance is multiplied by 1.02 each month. For example, after one month, the balance would be $1000 * 1.02 = $1020.
What is the pattern of shaded triangles in the Sierpinski Triangle, and how does it relate to the number of steps taken?
-In the Sierpinski Triangle, at each step, the number of shaded triangles increases by a factor of three, and the area of each triangle is divided by four. This pattern continues with each iteration.
How can you graph the number of shaded triangles and the area of each triangle in the Sierpinski Triangle as a function of the step number?
-You can graph the number of shaded triangles by plotting the count on the y-axis against the step number on the x-axis, showing an exponential increase. For the area of each triangle, plot the area on the y-axis against the step number, showing an exponential decrease.
What is the rule for creating a sequence where each number is four less than three times the previous number?
-To create a sequence where each number is four less than three times the previous number, you multiply the previous number by three and then subtract four to get the next number in the sequence.
How does changing the starting number in a sequence affect the subsequent terms when the rule is 'four less than three times the previous number'?
-Changing the starting number in the sequence will result in a different set of numbers following the same rule. Each subsequent term will still be calculated by multiplying the previous term by three and subtracting four, but the values will be different based on the initial number.
Can you provide an example of a different rule for generating a sequence, and what would the next three terms be if starting with 1?
-An example of a different rule could be 'multiply by -1'. Starting with 1, the next three terms would be: 1 * -1 = -1, -1 * -1 = 1, and 1 * -1 = -1.
Outlines
🔢 Understanding Geometric Sequences
This paragraph explains the concept of geometric sequences through examples. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The paragraph illustrates this with the sequence starting with 1 and a common ratio of 2, resulting in terms 1, 2, 4, 8, 16, and 32. It also discusses how to find the growth factor of a sequence, which is the multiplier used to get from one term to the next, using examples including a sequence decreasing from 256 to 128 with a growth factor of 1/2, and another increasing from 0.08 to 0.8 with a growth factor of 10. Additionally, it covers a practical example of a credit card balance increasing by 2% per month, showing how the balance grows using the growth factor of 1.02.
📊 Geometric Sequences in Geometry: The Cinsky Triangle
The second paragraph delves into a geometric application of sequences, specifically the Cinsky triangle problem. It starts with an equilateral triangle and iteratively breaks it into four smaller congruent triangles, removing the middle one. The process is repeated for each remaining triangle. The paragraph aims to complete a table showing the number of shaded triangles and their respective areas after each step. The number of shaded triangles increases by a factor of three with each step, while the area of each triangle decreases as the total area is divided among more triangles. This results in a pattern where the number of triangles grows exponentially, but the area per triangle decreases. The paragraph also includes instructions for graphing these sequences separately, showing how the number of shaded triangles increases while the area per triangle decreases with each step.
📐 Creating Custom Sequences with Mathematical Rules
The third paragraph introduces the creation of custom number sequences using specific rules. It presents a rule where each number is four less than three times the previous number, starting with 10, and builds a sequence of five numbers following this rule. The paragraph also encourages the selection of different starting numbers and the creation of sequences based on various rules, such as multiplying by -1 or subtracting two, to demonstrate the flexibility in sequence generation. This section showcases how different mathematical operations can be applied to create diverse sequences, emphasizing the importance of understanding the operations involved in sequence construction.
Mindmap
Keywords
💡Geometric Sequence
💡Growth Factor
💡Interest Rate
💡Cinski Triangle
💡Area
💡Shaded Triangles
💡Graph
💡Sequence
💡Rule
💡Decimal Place
Highlights
Geometric sequence is defined by a constant multiplication factor between terms.
The next three terms of a sequence are found by multiplying each term by the growth factor.
The growth factor for a sequence where each term is doubled is two.
To find the growth factor, divide the new term by the original term.
A sequence with a growth factor of one remains constant.
A sequence with a growth factor of 1/2 decreases by half with each term.
A sequence with a growth factor of 1/10 decreases by a factor of ten with each term.
A credit card balance with a 2% monthly interest rate increases by 102% of the previous month's balance.
The growth factor for a 2% monthly interest rate is 1.02.
The Sierpinski triangle is formed by repeatedly removing the middle triangle from a set of four.
The number of shaded triangles in the Sierpinski triangle pattern doubles with each step.
The area of each triangle in the Sierpinski pattern is one-fourth of the previous step's area.
The growth pattern of the Sierpinski triangle can be graphed to show the increasing number of triangles and decreasing area.
The number of shaded triangles in the Sierpinski triangle pattern increases exponentially with each step.
The area of each triangle in the Sierpinski pattern decreases as more triangles are formed.
A sequence where each number is four less than three times the previous number starts with a given initial value.
Different starting numbers yield different sequences following the same rule.
Alternative rules for sequence generation can be multiplication by a constant or subtraction by a constant.
Transcripts
number one says here are the first two
terms of a geometric sequence what are
the next three terms so remember that
this word geometric sequence means that
we are multiplying by a number okay so
there's multiplication happening here so
instead of addition it's going to be
multiplication so two to get to four
through multiplication is we multiply
each term by two so then it wants the
next um three
terms so we're going to multiply the
previous term time two so 4 * 2 is 8 8 *
2 is 16 and 16 * 2 is 32 so I'll just
write down what we did
here number two what is the growth
factor of each geometric sequence so the
growth factor is the number that you're
multiplying by to get the next term so 1
* 1 gives us 1 1 * 1 1 Time 1 1 Time 1
so our growth factor here is
one so what do we do to get from 256 to
128 okay so what are we multiplying by
and if you don't know okay you can
always take your new number divided by
your original number okay so the new
number number is 128 the original number
is
256 simplifies to 1/ 12 or5 so the
growth factor here is um
12 so what do we multiply 18 by to get
54 so again you can do 54 divided by 18
if you want to and that will give you
the growth factor of
three um in Part D we've got
0.08 ided
0.8 so this one has a growth factor of
um
1110th or um
0.1 okay now this one um the decimal
place we're getting um the number is
getting bigger each time by a decimal
place so the growth factor here is going
to be multiplying by 10 again you can
also do
0.08 divided by
0.008 and that would give you 10 as
well number three a PO a person owes um
$1,000 on a credit card that charges an
interest rate of 2% per month complete
this table showing the credit card
balance each month if they don't make
any
payments so um a growth rate of 2% per
month means they're going to owe 2% more
so they're going to owe their initial
100% plus they're going to owe an
additional 2% so each month they owe
102% of the previous month okay so we're
going to be multiplying and again you
could divide these to figure this out if
you didn't remember how to do that so
you could do um the growth factor as
1,00 20 divided 1,000 okay and this will
give you
1.02 which is 102% as a decimal so we're
just going to be multiplying um by 1.02
each
time and so when we multiply
1,40 and 400 by 1.02 we get 1,
16121 when we multiply this by
1.02 we get 1,
8243 next one we get
1,148 multiply by 1.02 again and you get
1,1
12616 multiply by 1.02 again and get
$1,148
69 so this is if they didn't make
payments and didn't get laate charges so
after that 8 months you already have
$148 in Interest being
charged number four we have this cinsky
triangle and that's where you start with
an equilateral triangle okay and then
you break it into four congruent
triangles and remove the middle one so
we can see that we've got four three are
shaded one is unshaded then you take
each of these and you do the same thing
so you can see here was this um
equilateral triangle split it into three
removed the middle here was this one
here was this one and so then that's
just going to continue happening so
complete the table showing the number of
shaded triangles in each step and the
area so we had one shaded triangle in
this first step we have one two three
shaded triangles in this next step and
now when we figure out the area we want
to think about how many triangles this
one was split into so this one was
256 so if we take
256 and we divide it by
four okay we're going to get the area of
one of those triangles which is
64 okay so it just wants one of the
equal triangles
there so now in this next one we have
three for each of these so each of these
triang each of these shaded triangles
gives us three more so now we're at
nine and now um we have four triangles
per here okay for each of these
triangles each one gets split into four
so where we had 1 2 3 four triangles now
we have um 16 we don't have it in here
okay but we would need to divide by how
many little triangles would fit in there
to get the area of each one so if we
were to draw in all these triangles
there would be 16 triangles so then
we're going to be doing 256 /
16 um which is
16 and you can kind of see the growth um
Factor happening here so this is just
divided by four okay this is divided by
four again so now we kind of have a
pattern to follow where this one is
multiplying by three and this one is
multiplying by three so now we'll know
each of these nine little shaded
triangles are going to produce three
more each so we'll do 9 * 3 which is
27 okay now we'll have to divide up
those triangles another four times so
the area is going to get divided by four
again so 16 divided by
4 then we have 27 * 3 so we'll get 8 81
here and we'll divide by four again and
get one multiply
81 by three again and get 243 little
triangles divide those each by four
again and we get 1 14 square
inch okay then Part B says to graph um
the number of shaded triangles as a
function of the step and then separately
graph the area of each triangle as a
function of the step number okay so
we're going to graph these each
separately um so I'm just going to kind
of sketch out these graphs here so in
this one we're going to go one through
five for the
steps and then we're going to go up to
243 so let me move this down a little
bit so I can extend this slightly so I'm
going to count um by let's see let's do
50s so this one will be 50 100 150 200
250 so in Step Zero we were at one so
I'm just graphing these so at Step Zero
we were at one so way down here since
this is 50 at step two we were at three
so just barely above step two we were at
nine step three we're at 27 which is
about halfway to
50 step step four we were at 81 okay so
not quite to 100
yet step five we were at 243 so not to
250 but pretty close so here's the graph
that this one is
creating so then we'll graph um the next
one so again we have steps 1 2 3 4 five
and this one is starting at um an area
of 256
okay so I'll count by 50s again so 50
100 150 200
250 so at Step
Zero our area was
256 okay so at zero we're at 256 so just
above
250 at 1 we were at 64 so here's
50 so we're at 64 so about here step two
we were at 16 so not even halfway to
50 step three we were at four step four
we were at one and step five we were at
1/4 so this one's
graph okay looks something like
this so how are they the same I would
say the same they both have
curves okay they because they're not
linear they're not changing at the same
rate but how are they different
different um the number of triangles is
going up where the area Okay so the area
per
triangle is going
down all right then number five says um
here is a rule to make a list of numbers
each number is four less than three
times the previous number so when you do
four less than
three times a
number okay you have to do the multiply
by three to the previous number first
then you subtract four so four less than
three times the number so in this first
one we're starting at 10 they want us to
build a sequence of five so we're going
to need four more
numbers so we're going to take 10 * 3
which is 30 then subtract
4 and we'll get
26 then we're going to
take um 26 *
3 and we'll
get
78 then we're going to subtract 4 which
is
74 then we'll do 74 *
3 which is
222 subtract 4
and we get
218
218 * 3 and we get
654 minus 4 gives us
650 then start with the number one and
build out a sequence of
five so um 1 * 3 is 3 - 4 gives us -1
-1 * 3 is
-3 - 4 is
-7 -7 * 3 is
-21 - 4 is
-25 -25 * 3 is
-75 - 4 is
-79 then it says just select a different
starting number and build a sequence of
five numbers so you can do whatever you
want here um you're just going to
multiply by three and then subtract four
so pick any number you feel
like number six a sequence starts Net
One negative 1 give a rule the sequence
could follow and the next three terms
okay so maybe you just want to multiply
so 1 * -
1 okay so we have one 1 and then we're
just going to multiply by1 for the next
three numbers so -1 * -1 is 1 1 * - 1 is
1 -1 * 1 is
1 then it says give a different rule
okay so maybe you just saw neg 2 to1 is
subtracting two okay so maybe for this
rule you're just going to subtract two
so 1 - 2 is
1 NE -1 - 2 is
-3 -3 - 2 is
-5 -5 minus 2 is -7 and obviously
there's multiple other rules you could
do for that that's just two examples
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