Grade 10 Math Q1 Ep5: Finding the Sum of the Terms of a Given Arithmetic Sequence
Summary
TLDRIn today's episode of 'adeptv', Sir Jason Flores guides viewers through the mathematical concepts of arithmetic sequences. The lesson focuses on calculating the sum of the first 'n' terms of such sequences, introducing formulas to simplify the process. Examples are provided to demonstrate how to use these formulas when both the first and last terms are known, or when only the first term and the common difference are given. The episode concludes with a real-world application, showing how these mathematical skills can be applied to saving money, as illustrated by a story about Jane saving for shoes.
Takeaways
- 📘 The lesson focuses on teaching the formula to find the sum of the first n terms of an arithmetic sequence, which is crucial for solving word problems involving series.
- 🔢 The formula for the sum of the first n terms when the first and last terms are known is S_n = (n/2)(a_1 + a_n).
- 🔄 When the last term is not given, an alternative formula is used: S_n = (n/2)(2a_1 + (n-1)d), where d is the common difference.
- 📝 An example is provided to demonstrate the calculation of the sum of the first 20 natural numbers, which equals 210.
- 💡 The lesson emphasizes the practicality of using formulas over manual addition for sequences with many terms, highlighting efficiency in computation.
- 📊 A step-by-step approach is shown for calculating the sum of terms in sequences, such as 5, 10, 15, ... up to 50, which sums to 275.
- 👟 A real-world application is presented where a student, Jane, saves money weekly, and the formula is used to calculate her total savings after 43 weeks.
- 🌟 The lesson concludes with a motivational message encouraging continuous practice and highlighting the relevance of math in daily life.
- 🎓 Sir Jason Flores, the presenter, aims to make learning math fun and easy, emphasizing the importance of logical reasoning and critical thinking skills.
Q & A
What is the main topic discussed in this episode?
-The episode focuses on finding the sum of the first n terms of an arithmetic sequence and solving word problems involving arithmetic series.
What formula is used to find the sum of the first n terms when both the first and last terms are given?
-The formula used is S(n) = n / 2 * (a1 + an), where S(n) is the sum, n is the number of terms, a1 is the first term, and an is the last term.
How do you find the sum of the first n terms when the last term is not given?
-When the last term is not given, the formula is S(n) = n / 2 * [2a1 + (n - 1)d], where d is the common difference between terms.
In the example, what is the sum of the first 20 natural numbers?
-The sum of the first 20 natural numbers is 210.
How is the sum of the sequence 5, 10, 15, 20, up to 50 calculated?
-By listing all the terms and adding them together, the sum is calculated as 275.
What is the result when calculating the sum of the first 16 terms of the sequence 8, 11, 14, 17, 20, etc.?
-The sum of the first 16 terms of this sequence is 488.
What is the formula used to solve the word problem about Jane saving money, and what is the final result?
-The formula used is S(n) = n / 2 * [2a1 + (n - 1)d], and after 43 deposits, Jane saves a total of 3,827 pesos.
What is the difference in the arithmetic sequence 1, 3, 5, 7, and so on?
-The common difference in this sequence is 2.
How is the sum of the sequence -3, -1, 1, 3, etc., calculated?
-Using the formula for when the last term is not given, the sum of the first 13 terms is 117.
What message does the episode conclude with regarding math?
-The episode encourages viewers to keep practicing math, emphasizing that math is part of daily life and can be fun and easy.
Outlines
📘 Introduction to Arithmetic Sequences
Sir Jason Flores introduces the lesson on arithmetic sequences, focusing on developing logical reasoning and critical thinking skills. The lesson aims to teach how to find the sum of the first n terms of an arithmetic sequence and solve related word problems. The importance of having learning modules, pens, and paper ready is emphasized. The previous lesson's discussion on finding the nth term of an arithmetic sequence using the formula a_n = a_1 + d(n-1) is recalled, and the new focus on the sum of the first n terms is introduced. Examples of sequences are given, and the impracticality of manual addition for long sequences is highlighted, leading to the introduction of a formula for faster computation.
🔢 Sum of Arithmetic Sequences Formulas
The video script explains two formulas for finding the sum of the first n terms of an arithmetic sequence. The first formula, S_n = (n/2)(a_1 + a_n), is used when the first and last terms are known. The second formula, S_n = (n/2)(2a_1 + (n-1)d), is used when the last term is not given. Examples are provided to illustrate the application of these formulas. The first example calculates the sum of the first 20 natural numbers, and the second example finds the sum of the first 16 terms of a sequence starting with 8 and having a common difference of 3. The process of substituting values into the formulas and performing the calculations is detailed.
📈 Practical Examples of Sum Calculations
This section presents practical examples to apply the learned formulas for calculating the sum of arithmetic sequences. The first example involves finding the sum of the first 30 terms of a sequence starting with 1 and having a common difference of 2. The formula S_n = (n/2)(2a_1 + (n-1)d) is used, and the calculation leads to a sum of 900. The second example calculates the sum of the first 10,000 terms of a sequence starting with 1 and ending with 10,000, using the formula S_n = (n/2)(a_1 + a_n), resulting in a sum of 50,005,000. These examples demonstrate the efficiency of the formulas in handling large sequences.
💼 Applying Arithmetic Sequences to Real-life Problems
The script transitions into applying arithmetic sequences to real-life scenarios with a word problem about Jane saving money for shoes. The problem describes an arithmetic sequence where Jane saves a weekly increasing amount starting from 5 pesos. The goal is to find out how much she saves after 43 weeks. The formula S_n = (n/2)(2a_1 + (n-1)d) is applied with the given values, leading to a total savings of 3,827 pesos. This example ties mathematical concepts to a relatable situation, showing how arithmetic sequences can be used in everyday life.
🎓 Summary and Conclusion
The final part of the script summarizes the key formulas for finding the sum of an arithmetic sequence: S_n = (n/2)(a_1 + a_n) when the first and last terms are given, and S_n = (n/2)(2a_1 + (n-1)d) when the last term is not given. A word problem involving Jane's savings is used to reinforce the learning, emphasizing the practical application of these formulas. The lesson concludes with an encouragement to practice and apply mathematical concepts in daily life, highlighting the fun and ease of learning math, and looking forward to the next episode.
Mindmap
Keywords
💡Arithmetic Sequence
💡Sum of the First n Terms
💡Common Difference (d)
💡nth Term (a_n)
💡Logical Reasoning
💡Critical Thinking Skills
💡Formula for Sum of an Arithmetic Sequence
💡Word Problems
💡Learning Module
💡Sequence
Highlights
Introduction to the episode focusing on arithmetic sequences and their sum.
Explanation of how to find the sum of the first n terms of an arithmetic sequence.
Practical example of manually adding terms of a sequence to find the sum.
Discussion on the impracticality of manual addition for sequences with many terms.
Presentation of the formula for the sum of an arithmetic sequence when the first and last terms are given.
Presentation of the formula for the sum when the last term is not given.
Example calculation of the sum of the first 20 natural numbers.
Example calculation of the sum of a sequence with a given first and last term.
Example calculation of the sum of a sequence without a given last term.
Exercise to find the sum of the first 30 terms of an arithmetic sequence.
Exercise to find the sum of a sequence up to ten thousand terms.
Exercise to find the sum of the first 13 terms of a sequence with a negative first term.
Word problem involving saving money in an arithmetic sequence pattern.
Solution to the word problem calculating total savings after 43 deposits.
Conclusion and encouragement for continued learning and practice in mathematics.
Transcripts
[Music]
hi
good day welcome in today's episode of
adeptv
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 yourself 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
first find the sum of the first
n terms of an arithmetic sequence
and second solve word problems
involving arithmetic series
in the previous episode it was discussed
that
to find the nth term of an arithmetic
sequence
the formula a sub n is equal to
a sub 1 plus d times
n minus 1 can be used
in this episode we will discuss how to
find
the sum of the first n terms of a given
arithmetic sequence
for example how do we compute the sum
of all the terms of each of the
following sequences
letter a 1 2
3 up to 100
and letter b 5 10
15 20 up to 50.
adding manually the terms of a sequence
is manageable
when there are only few terms in the
sequence
however if the sequence involves
numerous terms
then it is no longer practical to be
adding the terms
manually
it is a tedious work to do thus
this episode will present to you a
formula
that will make the computation faster
and easier to let you experience
getting the sum of the terms in a
sequence manually
let's do the following number one
find the sum of the first 20
natural numbers by listing
all the natural numbers from 1 to 20 and
adding them we have 1
plus 2 plus 3 plus 4
up to 20 that is equal to 210
thus the sum of the first 20
natural numbers is 210
number two find the sum of
all the terms of the sequence 5
10 15 20
up to 50. by listing all the terms of
the sequence
and adding them we have 5
plus 10 plus 15 plus
20 up to 50 you will get
275 thus
the sum of the terms of the sequence is
275
what if you're asked to find the sum of
the terms of the sequence
one two three
up to ten thousand there are ten
thousand terms to be added one by one
to get their sum right
in doing this kind of solution it is
very
challenging especially if you're dealing
with a sequence
that has many terms
in getting the sum of the terms of an
arithmetic sequence
we will be using any of the following
formula
s of n is equal to n
divided by 2 times a sub 1
plus a sub n if
the first and last term
are given again if the first
and last term are given we will use the
formula
s of n is equal to n divided by
2 times a sub 1 plus a sub
n where s sub n is
the sum of the first n terms
n corresponds to the nth position
a sub 1 is the first term
of the sequence and a sub n is
the last term
on the other hand if the last term is
not given
we will use the formula sub n
is equal to n divided by 2
times the quantity 2 times a sub 1
plus n minus 1 times d
again if the last term
is not given we will use the formula
s of n is equal to n divided by 2
times the quantity 2 times a sub 1
plus n minus 1 times d
where s sub n is the sum of the first
n terms a sub 1 is the first
term of the sequence and d
is the common difference
let's take a look at these examples
find the sum of the first 20
natural numbers from this sequence
we can see that our a sub 1 is equal to
1
our a sub n is equal to 20
our n is equal to 20 and we are looking
for s sub n since the first
and last terms are given we will use
the formula sub n
is equal to n divided by 2
times a sub 1 plus a sub n
substituting the given values in the
formula we will have
s sub 20
is equal to our n is 20
divided by 2 times
our a sub 1 is 1
plus our a sub n is equal to
20. next we will have
s sub 20 is equal to
20 divided by 2 will give us 10
and one plus 20 will give us
21. our s sub 20 will be the product of
10 and 21. we will get
210.
therefore the sum of the first
20 natural numbers is 210
now let's go to the next example
find the sum of the first 16 terms
of the arithmetic sequence 8
11 14 17
20 and so on
notice that in this sequence the given
are a sub 1 which is equal to 8 our n
is equal to 16. the difference
is 3 and we are looking for the sum
of the first 16 terms notice
that the last term is not given
so we will use the formula
sub n
is equal to n
divided by 2 times the quantity 2
times a sub 1 plus
n minus 1 times d
by substituting the given values in the
formula
we will have s
sub 16 is equal to
our n is 16
divided by two times the quantity two
our a sub one is eight
plus our n is
sixteen minus one
times the difference which is three
next we will have s sub 16
is equal to 16 divided by 2 is 8
times the quantity 2 times 8 will give
us
16 plus 16 minus 1
is 15 and the difference of
3.
next we will have s sub 16
is equal to 8 times the quantity sixteen
plus fifteen times three
will give us forty five
next we'll have s sub 16
is equal to 8 times the quantity of 16
plus 45 will give us
61.
our s of 16 is equal to the product of 8
and 61 that is equal to
488
thus the sum of the first 16 terms of
the series
is 488
great now let's move forward to the next
example
after knowing all the needed concepts in
finding the sum of an arithmetic
sequence
let's try to answer the following
exercises
find the sum of each arithmetic series
the first one find the sum
of the first 30 terms of the arithmetic
sequence
1 plus 3 plus
5 and so on
notice that in this problem our given a
sub one is equal to one
our n is thirty
our difference is 2 as you can see
by adding the difference to the
preceding term
to get the next term
and we are looking for s sub 30.
with this problem the last term
is not given so we will use
the formula s of n is equal to
n divided by 2 times the quantity
2 times a sub 1 plus
n minus 1 times d
substituting the values in the formula
we will get
s sub 30 is equal to
our n again is 30 divided by 2
times the quantity 2 times a sub 1 which
is 1
plus our n again is 30
minus 1 times the difference
2. next we have s sub 30
is equal to 30 divided by 2 will give us
15.
times the quantity 2 times 1 is 2
plus 30 minus 1 will give us
29 times the difference
2. moving forward we will have
s sub 30 is equal to 15
plus or the quantity 2
plus 29 times 2 is equal to
58 s sub 30
is equal to 15 times the quantity
of 2 plus 58 that's
60. our s
of 30 therefore is the product of
60 and 15 you will get
900.
thus the sum of the first 30 terms of
the sequence
is equal to 900.
excellent let's boost up your learning
with another example
find the sum of the sequence one
two three four
up to ten thousand
from the sequence we can notice that the
given
a sub one or first term is equal to one
our n is equal to ten thousand
our a sub n is equal to ten thousand
and we are looking for s sub n
also notice that the first
and the last term are given
so we will use the formula s sub n
is equal to n divided by 2
times a sub 1 plus a sub n
substitute the values in the given
formula we will have
s sub ten thousand
is equal to our n again is ten thousand
divided by two times
the first term which is one plus
our a sub n which is 10
000.
next we have s sub 10 000
is equal to 10 thousand divided by two
will give us five thousand
times the sum of one and ten thousand
you will have ten thousand one
our s sub 10 000 now is the product
of five thousand and ten thousand one
it will give us
this term therefore the
s sub 10 000 of the sequence
is 50 million 5
000. oh so
awesome fasten your seat belts as we
proceed
to the next exercise
find the sum of the first 13 terms
of the sequence negative 3
negative 1 1 3
and so on notice
that from the sequence our given a sub 1
or the first term
is negative 3. our n
is 13. the difference
is 2 that is by adding the common
difference
to the preceding term to get the next
term
and we are looking for s sub 13.
since the last term is not given
we will use the formula s of
n is equal to n
divided by 2 times
the quantity 2 times a sub 1 plus
n minus 1 times the difference
again substitute the values in the
formula
we will have s sub 13
is equal to n which is 13
divided by 2 times the quantity 2
times a sub 1 that is negative 3
plus our n which is
13 minus 1 times the difference
which is 2. next we have s
sub 13 is equal to 13 divided by 2
will give us correct 6.5 times the
quantity 2
times negative 3 this will be equal to
negative 6 plus
13 minus 1 will give us 12.
and times the difference which is 2.
moving on we have s sub 13
is equal to 6.5
times the quantity negative 6.
plus the product of 12 and 2
will give us 24.
next we have s sub 13
is equal to 6
point five times
get the sum of negative six and 24
will give us positive
eighteen that's correct
then our s sub 13 now will be equal to
the product of 6.5 and 18.
we will have 117.
so the sum of the 13 terms of this
sequence is equal to 117
how's that oh so amazing
well done congratulations
dear students on this part let us see
what you have learned from today's
episode
by answering the following questions
to find the sum of the terms of an
arithmetic sequence
what formula can be used if the first
term and last term are given
if the first term and last term are
given we use the formula
s sub n is equal to n divided by 2
times a sub 1 plus a sub n
what about if the last term is not given
what formula can we use
if the last term is not given we use the
formula
s sub n is equal to n divided by
2 times the quantity 2 times a sub 1
plus n minus 1 times d
now let's apply what you have learned by
solving
the given word problem
jane was saving for a pair of shoes
from her weekly allowance she was able
to save
five pesos on the first week
nine pesos on the second
13 pesos on the third week and so on
if she continued saving in this pattern
and made 43 deposits
how much did jane save
let's analyze and solve the problem
together
from the given problem we can see that
our first
term or a sub 1 is equal to 5 pesos
our n is equal to 43
the difference is equal to 4
and we're looking for s sub n
since the last term is not given
we will use the formula s sub n
is equal to n divided by 2 times
the quantity 2 times a sub 1 plus
n minus 1 times d
substituting the values in the formula
we will have
s sub 43 is equal to 43 divided by 2
times the quantity 2 times 5 plus
43 minus 1 times 4.
s sub 43 is equal to 43 divided by 2
is equal to 21.5
times the quantity 2 times 5 is 10
plus 43 minus 1 is 42
times 4 s of 43 is equal to 21.5
times the quantity 10 plus the product
of 42 and 4 that is 168.
s of 43 is equal to 21.5
times the sum of 10 and 168
that's one hundred seventy eight s sub
forty three
is the product of twenty one point five
and one hundred seventy eight that is
equal to three
thousand eight hundred twenty seven
thus after 43 deposits
jane can save 3827
pesos like
jane you too can save
start now your five pesos
can make a difference
oh so awesome right
i hope you learned a lot keep practicing
because
math is always part of our daily lives
[Music]
and that concludes our lesson for today
see you again on the next episode
and this has been sir jason flores also
bear in mind that learning math will
always be fun and easy
be also be awesome
only here on dappa tv
[Music]
you
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