GENERATING PATTERNS IN SERIES || GRADE 10 MATHEMATICS Q1
Summary
TLDRIn this video, Monica explains the concepts of finite and infinite series, and how to express a series using Sigma notation. She begins by defining a series as the sum of the terms of a sequence and provides examples of both finite and infinite series. Monica demonstrates how to calculate the sum of the first few terms in a sequence and introduces the concise Sigma notation for expressing series. The video includes detailed examples, helping viewers understand how to apply these mathematical concepts in various scenarios.
Takeaways
- 🔢 A series is the sum of the terms of a sequence.
- 🔍 A finite series, also called a partial sum, has a specific number of terms.
- ∞ An infinite series continues without end, summing terms indefinitely.
- 🧮 The sum of the first 'n' terms of a sequence is denoted by 'S_n'.
- 💡 Sigma notation (Σ) is used to express series concisely.
- 🔄 In Sigma notation, the Greek letter Σ represents summation, and the expression below it indicates the starting value.
- ➕ The upper limit of the summation is shown above the Σ symbol.
- 📏 Examples of sequences include 1, 3, 5, 7, 9, 11 and their sum is a series.
- 🔄 Finite series examples include the sum of terms like 1 + 3 + 6 + 10 + 15, which results in specific values.
- 🔍 Sigma notation can express complex series such as fractions or squared terms, simplifying the notation for summing sequences.
Q & A
What is a series?
-A series is the sum of the terms of a sequence.
What is the difference between a finite series and an infinite series?
-A finite series is the sum of a specific number of terms in a sequence, while an infinite series is the sum of all terms in an infinite sequence.
How is a series expressed in Sigma notation?
-A series can be expressed in Sigma notation by using the Greek letter Sigma (Σ) to indicate the summation of terms, specifying the lower and upper limits and the expression for the terms.
What is an example of converting a sequence to a series?
-If we have the sequence 1, 3, 5, 7, 9, 11, the series would be the sum of these terms: 1 + 3 + 5 + 7 + 9 + 11.
How do you calculate the sum of the first six terms of the sequence 1, 3, 5, 7, 9, 11?
-The sum of the first six terms is calculated as 1 + 3 + 5 + 7 + 9 + 11 = 36.
What is a partial sum in the context of series?
-A partial sum, or finite series, is the sum of the first n terms of a sequence.
How do you find the partial sum S_3 for the sequence 1, 3, 6, 10, 15?
-S_3 is the sum of the first three terms: 1 + 3 + 6 = 10.
What is the sum of the first four terms of the sequence 5, 15, 25, 35?
-The sum of the first four terms is 5 + 15 + 25 + 35 = 80.
How is the series 1 + 1/2 + 1/3 + 1/4 + 1/5 expressed in Sigma notation?
-The series is expressed as Σ (1/k) from k=1 to k=5.
How do you express the series 1^2 + 2^2 + 3^2 + ... + n^2 in Sigma notation?
-The series is expressed as Σ (k^2) from k=1 to k=n.
Outlines
📘 Introduction to Series
In this video, Monica Wama explains the concept of series, defining both finite and infinite series. A series is described as the sum of the terms of a sequence. An example sequence (1, 3, 5, 7, 9, 11) is provided, demonstrating how to form a series by summing these terms. The concept of a partial sum, such as S₆ (the sum of the first six terms), is introduced with examples showing how to calculate these sums. The difference between finite and infinite series is discussed, and various examples are given to illustrate these concepts.
📝 Sigma Notation
Sigma notation is introduced as a concise way to express series. The video explains that Sigma (Σ) is the Greek letter used to denote summation, instructing viewers on how to write series in this notation. Examples are provided to illustrate the process, such as expressing the series 3k (for k from 1 to 4) as Σ(3k) from k=1 to k=4. The video also addresses how to use different variables (i, j, k) for summation, with step-by-step examples showing how to substitute values and simplify the expression.
🔢 Complex Series in Sigma Notation
The video tackles more complex series, such as 1 + 1/2 + 1/3 + 1/4 + 1/5, and how to express them using Sigma notation. The pattern of numerators and denominators is identified, and the series is rewritten using summation. Another example is provided for the series 1² + 2² + 3² + ... + n², showing how to recognize patterns and apply Sigma notation. A complex alternating series (1, -3, 5, -7) is also covered, detailing the process of using Sigma notation to express the series with alternating signs and specific patterns in the terms.
Mindmap
Keywords
💡Series
💡Finite Series
💡Infinite Series
💡Sigma Notation
💡Partial Sum
💡Sequence
💡Summation
💡Terms
💡Lower Limit
💡Upper Limit
Highlights
Introduction to series, finite series, and infinite series.
Explanation of how a series is the sum of the terms of a sequence.
Example sequence: 1, 3, 5, 7, 9, 11.
Definition and example of finite series using partial sums.
Detailed steps for calculating partial sums of a sequence.
Example calculation of partial sums: 1 + 3 + 5 + 7 + 9 + 11.
Introduction to infinite series and its representation.
Distinction between finite series and infinite series.
Using Sigma notation to express a series concisely.
Explanation of Sigma notation as a Greek letter for summation.
Example of expressing a sequence using Sigma notation.
Step-by-step breakdown of converting a series to Sigma notation.
Example with Sigma notation: 1 + 1/2 + 1/3 + 1/4 + 1/5.
Using different variables in Sigma notation, such as i, j, or k.
Complex example of Sigma notation with alternating signs and polynomial terms.
Transcripts
[Music]
hi Monica wama in this video we will
define series finite series and infinite
series we will also express a series in
Sigma notation so first what is a series
a series is the sum of the terms of a
sequence so again I'm series ito yung
some nominal terms given an sequence so
Marin consequence begin add not
in China in teen atomic Nattie series so
halimbawa I have here 1 3 5 7 9 and 11
this is just a sequence pair of a pug in
AD not in Shaa Italian anti-natal Wagner
in series so young-sam yeah that is
there that is an example of series okay
I have here a sub six so since Y ni6 so
FX a be hand we will get the sum of the
first six terms so it's a one three five
seven nine at eleven kakuni not in
young-sam Neela since atonium Annina a
first six terms nothing off a bucket
first six terms kasam malaga ito i6 as S
sub six
alright an infinite series is the sum of
the terms of a sub 1 plus a sub 2 and so
on and so forth so forma Poppins in
nofap ignorant 'young infinite sequence
then madinat and a infinite series next
a finite series also called partial sum
so a tournament packs in a be nominating
partial sum or finite sea
Conan ad Newsome no sequence not enemy
room ending
okay so cappuccino Honiton young son and
finite sequence finite series and dog
not in done in the sequence 1 3 6 10 15
we have the following partial sums so
halimbawa i have a sub 1 so I'm first
I'm not in a 1 the first term of the
sequence a sub dweeb exhibition 1 plus 3
equals 4 y 1 plus 3 because this is the
sum of the first two terms as a 3 we
have 1 plus 3 plus 6 equals 10 because
this is the sum of the first three terms
so4 1 plus 3 plus 6 plus 10 is equal to
20 the sum of the first four terms and s
of 5 we have 1 plus 3 plus 6 plus 10
plus 15 is equal to 35 because that is
the sum of the first 5 terms so
depending on illuminance S sub 3 so
first three terms as sub 4 first four
terms another example as sub two ibig
sabihin
the sum of the first two terms on a
young fresh two terms not n we have 5
and 15 so therefore we will have pi plus
15 that is 20 s of 4 is equal to the sum
of the first four terms you Munim Appa
so S sub 4 we have 5 plus 15 plus 25
plus 35 and that is 80 another example
as a poor so we will get the sum of the
first four terms so we have
- negative 6:18 and negative 54 so we
will now have two plus negative six plus
18 plus negative 54 and that is equal to
negative 40 s sub 6 is the sum of the
first six terms so we will know how it's
a salmon anathan see 162 and negative
486 so we will have 2 plus negative 6
plus 18 plus negative 54 plus 160 2 plus
negative 486 and that is equal to
negative 364 so Gannon lamb
all right Sigma notation what is Sigma
notation a toy Emperor and unpack sulit
natin man series so since i'm gonnago
one attends a series I in ad not inches
and a hat Latin terms therefore we can
make use of summation notation or Sigma
notation
n-nobody tour this is the Greek letter
so parish and letter e nothing at about
nothing Sigma which tells us the sum or
add up the terms of a pagina gamete net
income symbol Neto shy inaccessible now
we need to add all the terms
okay so atrium concise way of expressing
a series so Anna battle open Abishag
inugami so the summation of three times
key starts with a taeyeon so a token at
o-18 tina tavad nadine index of
summation packs in a be nothing index
gentile maxi Simula orient in atomic
nothing lower limit so young kein Ayin
young young number come santa yo maxi
similar max substitute the once at a
time scale so halimbawa that is k is
equal to 1 so I'm gonna nothing gagawin
3 times 1 ok come sir anti Yamata tapas
Union
young Andaman Shyam and Gentile Matata
paws are you Tina tawa did nothing upper
limit so hidimba wa n is equal to for a
big Sabean Matata post is a three times
four so come on chemo is equal to 1 and
your n is equal to 4 therefore and McGee
ging Sammy summation mu I 3 times 1 plus
3 times 2 plus 3 times 3 plus 3 times 4
so meta kappa Spicer for Casilla n mo i
for okay now hindi l'm locking key and
inna gamut nothing symbol weather in
tango muhammad nam i weather in j so
fight a new letter pero a model a schema
gamete i I J or K ok let us try to
express each Psalm using Sigma notation
so give the new series and nothing shall
express using Sigma notation so
halimbawa marina hadith on 1 plus 1/2
plus 1/3 plus 1/4 plus 1/5 so Buffett
ganito and summation notation so again
ding-ding Yahoo under your pattern since
you given that in a series in a fraction
form therefore fraction form brain cha
ok now Buffett 1 and Allah gave us a
task as he put one on the pop and Cena
tense at a school when Eunice a
numerator now it on key bucket key
atonium key concern Malala Manhattan
cumson maxi see mullah Atkinson Matata
pause so Kayla delegate not engine oh
there one can say all of the numerators
I won UK since a by Bashar km ill allah
gave methane at veto not in Alana
Jenkinson maxi similar okay now we all
know that one item 1 and denominator
Union I won so epic Sabine's r1 shinnok
similar at NAPA Oh scheisse fine okay
so he didn't go so
is one soon Atlanta is a pattern since
the numerator not in a 1 and then UK
Gentile Maga assign come son Max is
Ebola at Matata pose so next touches a
one again chesa 5 so this is now our
summation notation for da given series
next so I have here 1 squared plus 2
squared plus three squared plus and
squared okay so I know gotta be nothing
so look at the pattern I'm partnering at
in so it okay gentlemen assign comes and
maxi similar Atkinson Matata pose so a
delegate not in K and then since my
square jaw so I'm simple not in yo and I
K squared okay honking at us and that
insha'Allah a guy
so since next in Malaysia so 1i n sy
Bacall Yamuna Sabah back on son come
axis Imola and then sunshine at a post
since wallet I am Number John ensure so
n and illaallah game at engine hey
so next arch is a 1 again chess again
this is our symbol next okay I have here
1 plus negative 3 plus 5 plus negative 7
ok so much of complicated data Noah
perros again it proved not and Buffett
at all so funny not in summation
notation attend at a summation notation
so next arches are one neg inches of war
so max a substitute is Imola 1 hung them
for cousin came at in a1 and you mess at
us not in eye for so let us pray to
substitute from 1 to 4 so a Tanisha
negative 1 raise to K plus 1 times 2 K
minus 1 so a substitute not in C 1 cos
ASA be dito ka is equal to 1 MUX type
note is a 1 so substitute nothing see
once among a key so Maggie game 1 plus 1
and adding exponent and then you took a
nut in Maggie
2 times 1 so let us all Maggie negative
1 raise to 2 and then 2 times 1 that
will become 2 and
and bring down minus one so we will have
negative one squared is positive 1 and
then 2 minus 1 that is also positive 1
so 1 times 1 is equal to 1 check not n
okay that is our first term next let's
try to substitute to NASA tuple entire
so my gigging yum K + 1 hat in McGinn 2
+ 1 u - k not in McGinn 2 times 2 so to
mnemonic is a substitute not in neon so
we will have negative 1 raised to 3 and
then 2 times 2 that will become 4 and
then minus 1 so negative 1 raised to 3
is still negative 1 and then 4 minus 1
that is 3 so therefore negative 1 times
3 that is negative 3 and this is our
second term ok next
so again since Sangam a purchase or nasa
triple entire so sorry substitute not in
see Theresa K + 1 and then 2 K Maggie
gain 2 times 3 so we will have negative
1 raise to 4 and then 2 times 3 we have
6 plus minus 1 so negative 1 raise to 4
is positive 1 and then 6 minus 1 is 5 so
therefore we will have 1 times 5 is
equal to 5 and that is our third term
next okay so at Onalaska say you knock
NASA some issue annotation a 10-4 unless
and more limit so you own NASA at the
ass so we will now have substitute not
in C 4 so 4 plus 1 an exponent not n and
then 2 times 4 so let us compute so 4
plus 1 that will become 5 and then 8 a 2
times 4 that is 8 so minus 1 therefore
we will have negative 1 raised to 5 that
is still negative 1 and then 8 minus 1
that is 7 so that is equal to negative 1
times 7
and that is negative seven and that is
the fourth term thank you for watching
this video I hope you learned something
don't forget to Like subscribe and hit
the bell button to our wall my channel
just keep on watching
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