Geometric Series | Finding the Sum of Geometric Sequence | Explain in Detailed |

Summary

In this video, Teacher MJ introduces geometric series, focusing on finding the sum of a geometric sequence. He demonstrates how to calculate the sum of the first five terms of a sequence manually and by using a formula. Teacher MJ walks through the steps for determining the common ratio and applying the formula, while also showing manual calculations to ensure accuracy. Additionally, he discusses special cases where manual computation is more efficient and challenges viewers to solve a problem themselves. The video offers an engaging math lesson, making complex concepts easier to understand.

Takeaways

• 📐 Geometric series refers to the sum of the terms in a geometric sequence.
• 🧮 Example problem: Find the sum of the first five terms in a sequence like 4, 12, 36, 108.
• ✍️ Manual addition works but is time-consuming, so a formula is preferred: Sₙ = a₁(1 - rⁿ) / (1 - r).
• 🔢 The common ratio (r) is found by dividing the second term by the first term, and similarly for other pairs of terms.
• 🔄 Using the formula, multiplying and simplifying the terms step-by-step gives the sum of the first five terms as 484.
• 🧩 If you calculate manually and compare with the formula, the results will match (e.g., adding terms directly also yields 484).
• ➗ In some cases, like repeating terms or alternating patterns, manual observation can show that the sum is zero without using the formula.
• 📝 When the common ratio is negative, extra care is needed with signs and powers.
• 💡 For sequences like alternating positive and negative terms (e.g., -3, 3, -3, 3), the sum of multiple terms may end up as zero.
• 🔍 For identical terms repeated over multiple terms, you can simply multiply the term by the number of terms to find the sum.

Q & A

What is a geometric series?

A geometric series refers to the sum of the terms in a geometric sequence, where each term is multiplied by a constant ratio from the previous term.

How do you find the common ratio in a geometric series?

To find the common ratio, divide the second term by the first term, and then divide the third term by the second term to ensure they are equal.

What is the formula to calculate the sum of a geometric series?

The formula for the sum of the first n terms of a geometric series is Sₙ = a₁ * (1 - rⁿ) / (1 - r), where a₁ is the first term, r is the common ratio, and n is the number of terms.

How is the sum of the first five terms of the sequence 4, 12, 36, 108 calculated using the formula?

First, identify the first term a₁ as 4, and the common ratio r as 3. Then apply the formula S₅ = 4 * (1 - 3⁵) / (1 - 3). Calculate 3⁵ = 243 and simplify the equation to get S₅ = 484.

How do you calculate the common ratio in the sequence 4, 12, 36, 108?

The common ratio is calculated by dividing 12 (the second term) by 4 (the first term), giving 3. To verify, divide 36 by 12, which also gives 3, confirming the common ratio is 3.

What is the common ratio and sum of the first six terms for the sequence 3, -6, 12, -24?

The common ratio is -2, calculated by dividing the second term (-6) by the first term (3). Using the formula, the sum of the first six terms is -63.

What happens when you multiply a negative number by itself an even number of times?

Multiplying a negative number by itself an even number of times results in a positive number, as the negative signs cancel each other out.

What is the result when adding the terms in the sequence -3, 3, -3, 3?

Adding the terms results in zero because each -3 cancels out with a positive 3.

When can you add the terms of a sequence manually without using the formula?

If the sequence follows a clear pattern, such as repeated pairs of opposite numbers that cancel each other out (like -3, 3, -3, 3), you can add the terms manually without using the formula.

How do you sum eight terms of the sequence 3/4, 3/4, 3/4, 3/4?

Since all terms are the same (3/4), add them directly. The sum is 3/4 * 8 = 6.

Outlines

00:00

📐 Introduction to Geometric Series

The teacher introduces the topic of geometric series, explaining that it refers to the sum of a geometric sequence. An example is provided where students are asked to find the sum of the first five terms of the sequence (4, 12, 36, 108). The teacher begins by demonstrating how to manually add the terms and explains that a formula, Sₙ, can be used to make the process easier. The components of the formula are explained: a₁ (the first term), r (the common ratio), and n (the number of terms).

05:01

📊 Applying the Geometric Series Formula

In this section, the teacher works through the solution to find the sum of the first five terms using the geometric series formula. They explain how to calculate the common ratio (r = 3) and raise it to the power of the number of terms. The teacher details the step-by-step process of calculating S₅, using 4 as the first term and r = 3. The result is S₅ = 484. The teacher then presents an alternative method to manually verify the result by adding the terms of the sequence, confirming that the sum is indeed 484.

10:03

🔢 Alternative Solution for Geometric Series

The teacher introduces a second way to solve the sum of a geometric series by using manual multiplication instead of dividing by the common ratio. This method involves calculating the product of terms and subtracting values. After demonstrating this method, the teacher confirms that the sum of the first five terms is still 484, reinforcing the reliability of both approaches to solving geometric series problems.

15:04

🔄 Exploring Symmetry in Geometric Sequences

In this section, the teacher discusses how in some sequences, you can find patterns that simplify solving the sum without using the formula. An example sequence alternating between negative and positive numbers shows that opposite terms cancel each other out, making the sum zero. The teacher walks through how to manually add these terms to arrive at zero and explains why this method is useful when the pattern of numbers is symmetric.

20:05

🧮 Manual Calculation of Fractional Sums

The teacher presents a new sequence involving fractions (3/4) repeated over eight terms and shows how to manually calculate the sum. Since all the terms are the same, the sum is found by multiplying the fraction by the number of terms. The teacher provides a quick refresher on how to add fractions with the same denominator and confirms that the sum is 6. This example illustrates how some problems can be solved quickly without complex formulas.

Keywords

💡Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the video, the teacher explains the concept and shows how to find the sum of the first five terms of a given sequence using a formula.

💡Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio. The video starts by describing a sequence with terms 4, 12, 36, and 108, where each term is multiplied by the common ratio of 3.

💡Common Ratio

The common ratio is the factor by which each term of a geometric sequence is multiplied to get the next term. In the video, the teacher demonstrates how to find the common ratio by dividing consecutive terms in the sequence, showing that the ratio is 3 in the example provided.

💡Sum of the First N Terms

This refers to the total of the first 'n' terms of a geometric sequence, found using a specific formula. The teacher in the video explains how to calculate this sum by substituting values into the formula, which involves the first term, the common ratio, and the number of terms.

💡Formula for Geometric Series

The formula for the sum of the first n terms of a geometric series is Sₙ = a₁(1 - rⁿ) / (1 - r), where a₁ is the first term, r is the common ratio, and n is the number of terms. In the video, this formula is introduced to simplify the process of finding the sum of the series.

💡Power of a Number

The power of a number refers to how many times that number is multiplied by itself. In the video, the teacher raises the common ratio, 3, to the power of 5 when calculating the sum of the first five terms of the sequence. This is demonstrated as 3 raised to the 5th power equals 243.

💡Negative Exponents

Negative exponents indicate that a number is being divided rather than multiplied. In the video, when calculating the sum of a geometric series with a common ratio of -2, the teacher demonstrates how to handle negative exponents and signs within the formula.

💡Manual Calculation

Manual calculation refers to solving a problem without using a formula, instead adding or multiplying numbers directly. The teacher contrasts manual calculation with formula use, showing how to manually add the terms of the sequence to verify the sum derived from the formula.

💡Zero Sum

A zero sum occurs when all terms of a sequence cancel each other out, resulting in a total of zero. In the video, the teacher provides an example where negative and positive terms in a sequence, such as -3 and 3, add up to zero, showing that the sum of certain sequences can be zero without needing a formula.

💡Negative and Positive Signs

Negative and positive signs play an important role in arithmetic, especially in sequences with alternating signs. In the video, the teacher emphasizes the correct handling of signs when multiplying or adding terms, particularly when explaining why negative times negative results in a positive value.

Highlights

Introduction to geometric series and its definition.

Explanation of the process of finding the sum of a geometric sequence manually.

Introducing the formula for geometric series sum, Sₙ = a₁(1 - rⁿ)/(1 - r).

Calculation example: Finding the sum of the first five terms of the sequence 4, 12, 36, 108.

Explanation of how to find the common ratio (r) by dividing the second term by the first term.

Demonstration of using the geometric sum formula with r = 3 and n = 5.

Detailed steps for calculating 3 raised to the power of 5 and simplifying the expression.

Alternative approach to solve the same problem manually for comparison.

The final sum of the first five terms: 484.

New example: Finding the sum of the first six terms of the sequence 6, -12, 24.

Explanation of how to deal with negative common ratios and applying the geometric sum formula with r = -2.

Manual calculation for the sequence: summing the terms gives -63 as the result.

Checking whether certain sums can be solved manually without a formula, like the sequence -3, 3, -3, 3.

Quick solution method for repeated terms where the sum cancels out to zero.

Final problem: Finding the sum of 8 terms where each term is ¾, resulting in a sum of 6.

Transcripts

hi guys good day this is teacher MJ and

our topic for Today class it's all about

geometric Series so without further Ado

let's do this topic so geometric series

class is you are referring to the sum of

the geometric sequence so example number

one find the sum of the first five terms

of this given sequence for 12 36 108 so

the thing that you will do class to find

the sum is you just simply add the

numbers 4 plus 12 plus 36 plus 108 now

since this is five terms you're still

lacking one more term to get the sum of

the five terms

so to do it manually you need you just

need to add this one four plus two plus

36 plus 108 plus another one more term

since this is five terms we only have

given four terms so you can actually do

it manually but it will take time that's

why we have the formula and the formula

that will be S Sub n so s of n this will

be the number of terms or the sum of the

terms that you are told to find so sum

of the first five terms so s of n

equals a sub 1 is the first term

quantity one minus r r is the common

ratio n will be the number of terms all

over 1 minus r

so let's try to solve this one using the

formula then later on we will do it

manually for you to really compare okay

so let's let's answer this one so S Sub

n that would be S Sub 5 the sum of the

first five terms a sub 1 is the first

term for

quantity one minus r is the common ratio

now to get the common ratio simply

divide the second term by the first term

so 12 divided by four that's three so

the common ratio is three you also need

to check plus the third term divided by

the second term so 36 divided by 12 is 3

so if they have the same answer

therefore the common ratio is three

that's how you get the comma ratio plus

simply divide the second term with the

first term and third term by the second

term to really check if they have the

same answer once their answers are the

same therefore the common ratio is

so 3 raised to the power of n will be

the number of terms that's five terms

you are told to find the sum of the

first five terms so you have five terms

for n

and one minus r is three

so this will be S Sub 5 the sum of the

first five terms

four times one minus three raised to the

power of 5 plus it means that you

multiply 3 by itself five times so there

would be three times three times three

times three times three

so three times three is nine nine times

three is twenty-seven

twenty seven times three that would be

81 81 times 3 that's 81 times 3 3 times

1 is 3 3 times 8 is 24.

243 so once again three times three

raised to the power of five it means you

multiply three by itself five times so

three times three times three times

three times three so that'll be 243.

all right so this will be 243

all over one minus three that's negative

two

so this will be S Sub 5 equals four

times you subtract this one simplify the

parenthesis one minus two four three

that's negative 242.

all over negative two

so in this scenario class you have two

solutions it's either you can multiply

this one

four times negative two four two and

then your answer you divide it by two or

you can simplify this one first okay the

fractions four divided by two since we

can divide four by two so in my case

class if I will be answering this one

I need to simplify this one first

and then the answer for this one is I

need to divide it by this one so that's

the thing that I will do it depends on U

plus you can do it this way 4 times

negative four two four two divided by

negative two or you can just simply

divide this one first to make the

numbers more okay is it it will be

easier for U plus to solve this one so

four divided by negative two that's

negative two because you will get small

numbers

so 4 divided by negative two that's

negative two then you multiply this one

negative two times negative two four two

okay so negative two

all right negative two times negative

two four two so let's ignore first the

signs of course negative times negative

it will be positive

right so negative

or Let's ignore the signs so two four

two times two because our answer here

here plus negative times negative it

will be positive so I just cancel this

out because 4 divided by negative 2 is

negative so 2 times 2 is 4

4 times 2 times 4 is 8 2 times 2 is 4

400

84. so that's the answer class S Sub 5

is equals to 484.

all right so that's the sum of the first

five terms now if you want to answer

this one with another solution so S Sub

5 let's try the other solution 4 times

negative two four two divided by

negative two so you can multiply that up

if you're confused with the individing

four divided by negative two go ahead

multiply that one so four

two four two

times four of course this is negative

negative two four two times four it will

be negative the answer will be negative

positive times negative is negative so 4

times 2 is 8 4 times 4 is 16 carry one

four times two is eight plus one is nine

so negative 968

and then divide it by negative two

of course your answer will be positive

because negative times negative is

positive so 968 divided by two so this

will be four four times two is eight

subtract nine minus eight is one bring

down six sixteen divided by two is eight

eight times three sixteen

subtract zero bring down eight

eight divided by two is four four times

two is eight

then zero

so this is positive because negative

divided by negative is positive so the

answer is four eight four you will get

the same answer because but for me it's

easy for you to answer this one if you

just divide four divided by negative two

okay so let's answer this one plus

manually let's check if we get the same

answer

so we still lock in one more term and

our common ratio is three

check that a while ago comment ratio is

three so multiply 108 by three

so eight times three is twenty four four

carry two three times zero is zero plus

two is two three times one is three so

three hundred twenty four will be the

next term

since this is five terms so one two

three four five so let's add up let's

add this up to check if we really get

484 so 324 plus 108

plus 36 so this will be plus 12 then

plus four

all right so let's add this up four plus

eight is twelve plus six is eighteen

plus two is twenty plus four is twenty

four four carry two

then two plus two is four plus three is

seven plus one is eight

three plus one is four four hundred

eighty four because so you get the same

answer right so 4. so you can do it

manually but it once again it will take

time so that's why we do have the

formula so our answer for number one is

484 right so this then the next term is

320 484.

all right so let me just right there S

Sub 5 is 4 8 4.

so let's try number two

there are some cases plus that you don't

need to use the formula you just look at

the terms and you can answer it right

away so let's try number two

so find the sum of the first six terms

of this given sequence

so six times three negative six twelve

negative twenty four

so first thing to do we substitute the

equation so this will be S Sub 6 we're

referring to the sum of the six terms

for six terms a sub 1 is 3

then one minus we get the common ratio

so second term divided by the first term

so negative six

divided by three so there will be

negative two then you check you also

divide third term by the second term so

twelve divided by negative six so

positive divided by negative is negative

two so the common ratio is negative

all right

so this will be R is negative two you

put parenthesis class since we already

have assigned the minus sign so we're

not allowed that the signs are close to

each other so therefore we put

parenthesis so this will be negative two

then parentheses our n is the number of

terms six terms

all right sixth terms so the N is six

then close parenthesis all over 1 minus

our R is negative two so you put

parenthesis then negative

so this will be

three so sum of the sixth terms so once

again you put parentheses class because

we already have the minus one one minus

the the equation plus okay the formula

one minus then your R is negative two

that's why you put parenthesis because

we're not allowed to have two negatives

close to each other

so that's why we need to put parenthesis

so this will be 1 minus so two negative

two raised to the power of six it means

that you multiply negative two by itself

six times so negative two times negative

two

that'll be positive four times negative

two that's negative eight so we only we

already have three twos one two three

so this will be negative eight times

negative two negative times negative is

positive sixteen

times negative two this will be sixteen

times negative two that's negative 32

so we only have one two how many twos

now one two three four five

last one negative

negative 32 times negative two the

answer is negative times negative which

would be positive 64.

all right so this will be positive 64.

and this one this will be

one minus

so this one class is you need to

multiply the signs

right so negative times negative and

this will be positive plus

okay the sign will be positive because

once again do not multiply 1 times 2 you

only multiply the size because we have

two signs here so negative times

negative is positive then capital

so where S Sub 6 equals three times one

minus 64. that's negative 63 all over or

close parenthesis then divide 1 plus 2

is 3.

all right and then you can cancel this

out

so cancel three

so the remaining will be one

or negative 63 1 times negative 63 is

negative 63. so you cancel this out and

their meaning will this would be this

negative 63. so the sum okay so the sum

S Sub 6 is equals to negative 63. that's

the answer plus for number two

all right because we can divide three by

three so one plus two is three so three

divided by three is one one times

negative 63 is negative 63. so that's

the answer for number two easy for

number two right quite complicated

because the ratio is negative but that's

the thing that you will do

all right so let's try number three now

for number three and four class

you don't need to use the formula why is

that sir so it's because for number

three if you're getting the sum of the

six terms you try to check class if you

will be adding this one

okay if you will be adding this one

negative three plus three this is zero

so negative three plus three this will

be zero then the other one negative

three plus three this will be zero so

zero plus zero okay let me explain that

for number six

for number six

and for number four

okay the teacher will give you this kind

of example you don't need to use the

formula because you can just simply okay

simply look look at this one and then

you get the sum example number six

negative three three

negative three three

negative three three so once again

you're referring to the sum so if you

add this one negative three this one

negative three plus three this will be

zero right this is zero negative three

plus three this is cancel this is zero

and then you add this another you add

this another

three three negative three and three

negative three plus another three this

will be zero then negative three plus

another three this will be zero so zero

plus zero

plus zero okay this will be zero so the

S Sub six or the sum

of the six terms of this given sequence

is zero

because if you do it manually class you

will get zero so negative three plus

three equals zero plus negative three

plus three okay this will be the case

negative three plus three

plus another negative three

plus three

plus another negative three plus three

so you can cancel this out you can

cancel this out because negative three

plus three that's zero and you can

cancel this out so not using the

equation plus you know that the sum will

be zero

so that's how you answer class number

three

number three and four you check plus the

the sequence

okay so the answer for number three is

sub six

so number two is S Sub six is negative

63 a while ago and S Sub 6 where number

three is zero even if we use the formula

okay let's try to use the formula

okay S Sub n so S Sub 6 we're number

three first term is negative three times

one minus r is the common ratio

so common ratio is so you divide plus

three divided by negative three

so three divided by negative three

that's negative one you also check the

third term by the second divide by the

second term negative three divided by

three

it's negative one so the common ratio is

negative one you put parenthesis the

negative one then n is six then

close parenthesis

so the common ratio is negative one

once again you always need to put

parenthesis because our common relation

is negative one and you already have a

negative sign

before negative we already have in the

equation class you already have the

formula you already have the negative

and your common ratio is negative one so

that's why we need to put parenthesis

because we're not allowed to have two

signs close to each other

and then 1 minus the common ratio is

negative one

so this will be S Sub 6 let's check if

we get zero

negative 3

times this is 1 minus negative 1 raised

to the power of six this is negative one

times negative one this will be positive

one so you multiply negative one by

itself six times and you will answer

will be positive one because negative

times negative is positive

so we have two even negatives so

negative times negative is positive

another two negatives so this is example

this is one one negative one negative

one negative one and negative one and

another negative one a negative one so

therefore this should be positive

negative times negative is positive

negative times negative is positive and

negative times negative is positive but

if you're confused with this one class

go ahead you can multiply this one six

times negative one times negative one

that's positive one

times negative one so we only already

have two sorry three negative ones one

times negative one that's negative one

times

another negative one

so we already have four negative one

times negative one that's positive one

times negative one so fifth negative one

that would be negative one

times negative one and the answer is

positive one

so the answer is positive one

then

this will be

all over 1 minus this will be 1 minus

negative one so you multiply the sign so

negative times negative this will be

positive one plus one so S Sub 6 equals

negative three times one minus one is

zero

all over one plus one is two

so S Sub 6 equals negative three times

zero that is zero any number multiplied

by zero the answer is zero so zero

divided by two is equals to zero so

that's why our answer a while ago is

zero so you can do it manually class

it will take time if you do it this way

by just checking at the sequence you

already know that the answer is zero

same with number four class

stray number four

so if you check this one glass let's

let's just do it manually class because

there are some cases that the examples

are can be answered by doing it manually

by just check by just looking at the

sequence you already know that you can

answer this up by just looking by just

doing it manually

so for number

four class find the sum of the eight

terms first eight terms of this given

sequence we already know that if we add

this one three over four plus three over

four plus three over four plus three

over four since we have eight terms and

the numbers are just the same

so plus three over four

so plus three over four so we have one

two three four five six seven and eight

now in Productions class if they have

the same denominators if you will be

adding these fractions of course the

next number will be three over four T

over four

because the common ratio plus is just

one okay one times three over four

that's three over four three over four

times one that's three over four so

that's why you have three over four the

same numbers

so if you add this one out very easy for

the rules of fractions simply copy the

denominator

so copy four and then add the numerator

so three plus three

is six plus three is nine plus three is

twelve plus three is fifteen plus three

is eighteen plus three is twenty one

plus three is twenty four or you can

multiply three times eight because we

have three eighths three times eight is

twenty four then divide 24 divided by

four that is six

so for number four class the sum of the

first eight terms that would be six

that's that's it that's for number four

S Sub six is

another it's S Sub 8 sum of the first

eight terms that is six

so there are some cases plus that you

you will not use the formula because you

can do it manually

so you try number five glass and you put

your answer in the comment section down

below

okay you try number five and you put

your answer in the comment section down

below I will be checking that one

so once again if you want to know the

solution of number four feel free to

leave a comment on the section in the

comment section down below because I

will be solving the equation or the

solution for number four

so try to answer number five and I hope

you learned something new today so if

you like this video do not forget to

subscribe you share it to your friend's

class and your classmates so that we can

help more students more people so that

math would be easy as it could

so once again this is teacher MJ and you

have a great day goodbye for now bye bye