PROBLEM SOLVING WITH PATTERNS || MATHEMATICS IN THE MODERN WORLD

Summary

This video discusses problem-solving techniques using patterns and sequences, focusing on concepts like difference tables and Fibonacci numbers. It explains how to predict the next term in a sequence by analyzing differences and how to find the general nth term formula for arithmetic sequences. The video also covers the Fibonacci sequence, its properties, and methods for calculating specific terms using Binet’s formula. With practical examples and step-by-step demonstrations, the video helps viewers understand sequence patterns and apply mathematical formulas to solve problems.

Takeaways

• 📐 A sequence is an ordered list of numbers where each number is called a term, denoted as a_n for the nth term.
• 🔢 To find the next term in a sequence, the difference table method can be used by calculating first and second differences.
• 📈 If second differences are equal, it indicates a pattern that can be used to predict the next term in the sequence.
• 📋 The formula for the nth term of a sequence can be derived from patterns, such as a_n = n^2 + n.
• 🔑 Arithmetic sequences have a common difference, denoted as 'd', which can be used to find any term using the formula a_n = a_1 + (n-1) × d.
• 🧩 The pattern of square tiles can be used to determine the nth term formula, such as a_n = 3n - 1 for a specific pattern of tiles.
• 🌐 The Fibonacci sequence is a series where each number is the sum of the two preceding ones, represented by the formula f_n = f_{n-1} + f_{n-2}.
• 📉 Binet's formula is a direct way to calculate Fibonacci numbers without using a sequence table, involving the golden ratio.
• 🔄 A property of Fibonacci numbers is that for n ≥ 3, 3 × f_{n-1} - f_{n-2} = f_{n+1}.
• ❌ The statement that if n is even, then f_n is even, is false because the pattern of even and odd Fibonacci numbers alternates.

Q & A

What is a sequence in mathematics?

A sequence is an ordered list of numbers, where each number is called a term. It is a function whose domain is the set of positive integers.

How is the nth term of a sequence denoted?

The nth term of a sequence is denoted as a sub n (aₙ). For example, a₁ is the first term, a₂ is the second term, and so on.

What method is used in the video to predict the next term of a sequence?

The difference table method is used to predict the next term of a sequence. By finding the first and second differences, if the differences become equal, the next term can be predicted.

How do you calculate the first differences in a sequence?

The first differences are calculated by subtracting each term from the next. For example, in the sequence 5, 14, 27, 44, 65, the first differences are calculated as 14 - 5 = 9, 27 - 14 = 13, 44 - 27 = 17, and so on.

How are the second differences calculated, and what does it signify?

The second differences are calculated by subtracting the first differences from each other. For example, 13 - 9 = 4, 17 - 13 = 4, and so on. If the second differences are equal, it indicates that a pattern can be used to predict the next term.

What is the nth term formula mentioned in the video?

An nth term formula is a mathematical expression that generates the terms of a sequence. For example, the formula aₙ = 3n² + n can be used to find terms in a quadratic sequence.

How is the common difference used to find the nth term of an arithmetic sequence?

In an arithmetic sequence, the common difference is used in the formula aₙ = a₁ + (n - 1) × d, where a₁ is the first term and d is the common difference. This formula helps calculate any term in the sequence.

What is the Fibonacci sequence, and how is it generated?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The formula to generate the Fibonacci sequence is Fₙ = Fₙ₋₁ + Fₙ₋₂, where Fₙ represents the nth term.

What is Binet’s formula for calculating Fibonacci numbers?

Binet's formula is a closed-form expression used to calculate Fibonacci numbers directly. It incorporates the golden ratio and is expressed as: Fₙ = (1/√5) × [(1 + √5)/2]^n - [(1 - √5)/2]^n.

How is the difference table method different from Binet's formula when predicting terms in a sequence?

The difference table method predicts the next term by using differences between terms and relies on identifying a pattern in those differences. Binet’s formula, on the other hand, is a direct mathematical formula used specifically for Fibonacci numbers, which avoids the need to calculate prior terms manually.

Outlines

00:00

🔢 Understanding Sequences and Difference Tables

The video introduces the concept of sequences, an ordered list of numbers, and their terms. A method to predict the next term of a sequence using a difference table is explained. First, the differences between consecutive terms are calculated, followed by the second differences. When these become constant, it helps predict future terms. The video demonstrates this with a sequence of numbers and walks through the process of calculating the next term (90) by adding up first and second differences.

05:03

📊 Predicting the Next Term Using Difference Tables

This section continues with the difference table method for predicting the next term of a sequence. The example starts with the numbers 7, 24, 59, 118, 207, 332, and the first, second, and third differences are calculated. Since the third differences are constant (all equal to 6), the next term can be predicted. Using this method, the next number in the sequence is found to be 499. The explanation emphasizes the steps involved in calculating the differences and adding them back to find the next term.

10:28

📐 Generating Terms from a Sequence Formula

The focus shifts to using a formula to generate the terms of a sequence. A specific sequence formula, a_n = n^2 + n, is provided to find the first five terms of a sequence. The formula is applied by substituting values of n into the equation, resulting in terms such as 4, 14, 30, 52, and 80. Additionally, the process for determining the nth term of an arithmetic sequence using common differences is explained, and examples with sequences and their common differences are provided.

15:28

➗ Applying Arithmetic Sequence Formula

The video illustrates how to apply the arithmetic sequence formula, a_n = a_1 + (n-1)d, to calculate terms in a sequence. Given a common difference of 6 and a first term of 4, the formula is used to calculate the nth term of a sequence (4, 10, 16, 22, 28). The method of distributing and simplifying the terms in the equation is discussed, leading to the general form 6n - 2 for the sequence. A practical example with square tiles is introduced to demonstrate how to count tiles in geometric patterns using this approach.

20:47

🔢 Fibonacci Sequence and Pattern Formulas

This section introduces the Fibonacci sequence, a series where each term is the sum of the two preceding terms. The formula for Fibonacci numbers, F_n = F_{n-1} + F_{n-2}, is explained, and examples of calculating specific Fibonacci terms (e.g., the 10th and 13th terms) are given. The video also explores an alternative method for calculating Fibonacci numbers using Binet’s formula, which incorporates the golden ratio. A calculator-based demonstration shows how to efficiently find Fibonacci terms.

🧮 Exploring Properties of Fibonacci Numbers

The video examines the properties of Fibonacci numbers. For instance, the statement 3F_n - F_{n-2} = F_{n+2} is analyzed and shown to be true for n >= 3. Additionally, the claim that Fibonacci numbers are odd when n is even is evaluated and found to be false. The segment wraps up with a discussion on how to use a calculator to quickly solve Fibonacci-related problems and verify the properties of these numbers.

Keywords

💡Sequence

A sequence is an ordered list of numbers in which each number is referred to as a term. In the video, the sequence is introduced as a fundamental concept in problem solving with patterns. The example of a sequence given in the video is 5, 14, 27, 44, and 65, and the challenge is to predict the next term.

💡Term

A term refers to each individual number in a sequence. In the video, the presenter explains how each term is labeled as 'a sub n,' where 'n' represents the position of the number in the sequence. For example, the first term in the sequence 5, 14, 27, 44, 65 is 5, which is denoted as 'a sub 1.'

💡Difference table

The difference table is a method used to find the next term in a sequence by calculating the differences between consecutive terms. In the video, the first and second differences of the sequence are calculated to show how the differences become equal, allowing for prediction of the next term.

💡First difference

First differences are the differences obtained by subtracting each term in a sequence from the following term. In the video, for the sequence 5, 14, 27, 44, 65, the first differences are 9, 13, 17, and 21. These differences help identify patterns within the sequence.

💡Second difference

The second difference refers to the differences between the first differences. In the video, the second differences for the sequence 5, 14, 27, 44, 65 are calculated as 4, 4, and 4, indicating that the sequence follows a predictable quadratic pattern.

💡Nth term

The nth term represents the general form of a term in a sequence, allowing one to predict any term's value. In the video, the presenter introduces the nth term formula for sequences, such as 'a sub n = n squared + n,' and demonstrates how it can be used to find specific terms in the sequence.

💡Arithmetic sequence

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. In the video, an example of an arithmetic sequence is given with a common difference of 6. The formula 'a sub n = a sub 1 + (n-1)d' is used to find the nth term.

💡Fibonacci sequence

The Fibonacci sequence is a famous pattern where each term is the sum of the two preceding terms. In the video, the presenter discusses the first 14 terms of the Fibonacci sequence and explains how the Fibonacci formula can be used to calculate any term, using both the recursive method and Binet’s formula.

💡Binet's formula

Binet's formula is an explicit formula used to calculate Fibonacci numbers without recursion. It incorporates the golden ratio and allows for quick calculation of any Fibonacci number. In the video, the presenter explains how this formula can be applied using a calculator to find specific Fibonacci terms, such as the 13th term.

💡Common difference

The common difference is the consistent difference between consecutive terms in an arithmetic sequence. In the video, the common difference is found to be 6 in a sequence with terms like 4, 10, 16, 22, and 28, helping to derive the formula for the nth term.

Highlights

Introduction to sequences and their representation as functions.

Explanation of terms within a sequence and their notation.

The concept of predicting the next term in a sequence using a difference table method.

Demonstration of calculating first differences to identify patterns.

Procedure for finding second differences when first differences are not equal.

Prediction of the next term in a sequence using equal second differences.

Example of predicting the next term in the sequence 5, 14, 27, 44, 65.

Application of the difference table method to a different sequence.

Explanation of constructing a formula for the nth term of a sequence.

Derivation of the first five terms using the formula 3n^2 + n.

Introduction to arithmetic sequences and their general formula.

Finding the nth term of an arithmetic sequence using the formula a_n = a_1 + (n-1)d.

Example of calculating the number of tiles in a pattern using the nth term formula.

Introduction to the Fibonacci sequence and its properties.

Explanation of finding a term in the Fibonacci sequence using the recursive formula.

Use of Binet's formula to calculate Fibonacci numbers without a table.

Properties of Fibonacci numbers, such as the relationship between 3 times a term and the sum of two other terms.

Discussion on the parity of Fibonacci numbers and its implications.

Conclusion and call to action for viewers to engage with the channel.

Transcripts

hello everyone so in this video we are

going to discuss

problem solving with patterns

an ordered list of numbers such as 5 14

27 44 65

we call that as a sequence a sequence is

a function

whose domain is the set of positive

integers

it also means an ordered list of numbers

each number

in a sequence is called a term

5 14 27 44 65 we need note the term

as follows so a sub 1 so that is our

first term

our second term is 14 we denote as a sub

2

and a sub 3 we denote that is for our

third term

27 and then a sub 4 is equal to 44 that

is our fourth term

and a sub 5 is equal to 65 and that is

our

fifth term so we call this a sub

n is the nth term of a sequence now

as you can see we have a tree that's

here

so it indicates that the sequence

continues

beyond 65

what is the next term in the sequence 5

14

27 44 and 65

so to answer these questions we often

construct a different stable

so we are going to use this method so we

have a

different way to get the next term in

the given sequence

so in this time we are using the

difference table method

so by using the sequence we are going to

get the differences

wherein if the differences are already

equal

we can now predict the next term how to

do that

okay first we are going to get first

the first differences so pannu

we are going to subtract the first term

from the second term so 14

minus 5 that is 9 27

minus 14 that is 13 44 minus 27 that is

17

65 minus 17 that is

a 65 minus 44 that is 21 as you can see

class

so in the first differences the

difference are not yet equal so

we're going to proceed for the next

differences so that

is our second differences so 13 minus 9

that is 4 17 minus 13 that is 4

21 minus 17 that is 4. okay as you can

see

uh in our second differences uh equal

necessary so therefore we can predict

the next

uh number here in the first differences

so e big sub b hand the next term here

if you subtract the next term here to 21

the difference also is 20 uh four

so therefore on gagovindatan we're going

to add four

here so 21 plus four so the next term in

the first differences

in this uh row is 25

and then we add upwards so panning

next term details so for so for example

here so par paul the coin 65 because 44

plus 21 that is 65 second indian d2

process 65 plus 21

25 that is 90 so therefore

the next term in the given sequence by

14 27 44 65

is 90. so i know it again i mean nothing

metadata excuse difference table

next

we have 27 24 59 118

207 and 332

so we're going to predict the next term

so first we're going to get the first

differences so

7 minus 2 that is 5 24 minus 7 that is

17

and 59 minus 24 that is 35

118 minus 59 that is 59 also

and then 207 minus 118 that is 89

332 minus 207 that is 125.

so as you can see in our first

differences

in the indeep equal no your differences

so

let's proceed for the second differences

17 minus 5 that is 12 35 minus 17 that

is 18

59 minus 35 that is 24

89 minus 59 that is 30 125 minus 89 that

is 36 so

against the second difference says

nothing hindi pencil

equals so therefore let's proceed for

the third differences

so 18 minus 12 that is 6 24 minus 18

that is 6

30 minus 24 that is 6 36 minus 30 that

is 6 now

so my kitten at n they are equal now so

we can

get the next term in this row so one

uh differences this is six so we can add

six

to get this uh the next term so 36 plus

six that

is 42 and then targeting detail we're

going to add

okay and the last term and the first

differences so 125 parama

na tinto 125 plus 42 that

is hundred sixty-seven parameters a

given sequence nothing

so we're going to add 332 plus 167

and that is 499. okay so you know how to

predict

predict the next number

okay in the previous example

in the previous example we use

difference table to predict the next

term

of a sequence so in some cases we can

use

pattern to predict a formula that we

call the n term formula

so that generates the term of a sequence

so for example we have

a sub n is equal to n squared plus n so

we're going to

find out now we are going to find the

first five terms

so given this rule okay so

we're going to substitute so a sub 1

because we denote that as the first term

so

1 so that's why 3 times 1 squared plus

1 and then it's equal to 1 times 1

because 1 squared is 1 so that is 3

times one

plus one and that is equals to four so

three times one is three plus one that

is four so eb sub

and your first term not n is four next

we have a sub two so the second term

we're going to get the second term

so that is 2 squared that is 4 so 4

times 3

12 plus 2 that is equal to 14. so

therefore

the second term is 14 next

the third term so the third term we

substitute three

so salihatnam values so three squared

plus three and three squared is nine

so nine times three this is equal to 27

plus three

and that is thirty so it is a b and

thirty young

third term not ten and for the fourth

term we have four

squared that is sixteen times three

so sixteen times three plus four that is

52 and last the

fifth term so we have we are going to

substitute 5

so n not n so 5 squared that is 25

times 3 plus 5 is equal to

80 so so the first five terms of the

sequence

given the n term 3 n squared plus n are

4 14 30 52 and 80.

sequence okay what is the end term of

the sequence

4 10 16 22 and 28 so

nothing young and term in the previous

slide

bini gaya young and term okay now

gamet's young given sequence kuni

nothing

and term okay so gamete

general rule for n terms sorry to

lagging in a gamut

so we're going to get the first hour

just

substitute know what nothing

says nothing so in a sub 1 d this is the

first term

young and not in detail that is the uh

number position in the given sequence

and d is the common difference so

pakistan be nothing common difference

you know is nothing silly something

common difference so for example 10

minus 4

that is 6 16 minus 10 6 by 10

22 minus 16 that is 6 28 minus 22 that

is 6. so therefore

uncommon difference d to

so in short you know a sub one nothing

is four because that is the first term

and then you d not end which is the

common difference that is six again

but you know common difference so emma

minus like not

m 10 minus 4 that is 6 16 minus 10

that is 6. so please check then you know

given sequence nothing or arithmetic

yeah

sometimes uh on the first and second

10 16 22 and 28.

okay so substitute a sub 1 that is 4

plus n minus 1 and the common difference

is 6

and then simplify so multiply not n and

6 by using distributive properties so

six times

n that is six n six times negative one

that is negative six and then combine

similar terms

since c six and one capacitance

so copy six n and let's see four minus

six

ang paksa my name so four minus six that

is negative two therefore

the nth term of the sequence 4 10 16

22 and 28 is 6 n

minus 2

another assume the pattern shown by the

square

tile in the following figure continues

so this is our first term so

meron dalawang

eight so we have one two three four five

six seven eight nine ten eleven

so meron thailand eleven a square tile

so

allah me nothing

so what is the first term that is two

okay

so in common difference nila so

first term y two and this is five five

minus two three

ito i eight so eight minus five three

then so it is 11 11 minus eight that is

three

so therefore uncommon difference detail

i3

now substituting on the

uh in this formula so we come up so

a sub one nothing two plus n minus one

times three and then distributing three

inside the parentheses so they have we

have

two plus two three times n that is three

n

minus three and that is three n minus

one bucket

simplify so 2 minus 3 that is negative 1

so therefore

an n term detail is 3 n minus 1. next

question

how many tiles are in the eight figures

of the sequence so

instead uh uh draw kajan i'm a square

tile so why not

using the enter mao kuan

substitute in the given n term so we

have three n minus one so

papalitan and so 3 times 8

24 minus 1 that is 23 so therefore

i'm putting 8 figures now i met on 23

square

tiles another which figure will

consist of exactly 320 tiles so

pang ilan kaya in figure nao na meron

320 tiles

so using this and term so

a big sub hand so allah

320 tiles so three n

minus one so it is

so 3n minus 1 is equal to 320 transpose

negative 1 so that will become 320 plus

1.

so 3n is equal to 321 divide both sides

by 3 so you

figures okay next

fibonacci sequence this is included in

the problem solving

with patterns so when you say fibonacci

fibonacci sequence that is

the series of numbers such that the next

number

is found by adding up the two numbers

before

it so we can use this formula f sub n is

equal to

f sub n minus 1 plus f

sub n minus 2 where n f

sub n is the term number n

or this the number position on the uh

given term next f sub n

minus 1 is the previous term or n minus

1

for f sub n minus 2 is the term before

that

n minus 2. now

okay for example so this is the first

14 terms now this is the first 14 terms

in the fibonacci sequence

okay so let's try find the 10th term in

the

fibonacci sequence so using the formula

so f sub n

is equal to f sub n minus one

plus f sub n minus two so

so that will become ten minus one that

is

f sub nine d to the n minus two that is

f sub

eight so nothing f sub nine say it

on now a new volume

f sub n yeah that is thirty four you

know f sub 8 amount that

is 21 so 34 plus 28 that is 55 it makes

a bn

so this is the first tenth term

fibonacci sequence nothing or so

fibonacci numbers nothing so

young f sub 10 that is equal to 55

another

find the 13th term in the fibonacci

sequence so

we have okay using the formulas

and 13 so 13 minus 1 that is 12

plus so d to 11 so in value number 12

nothing that is 144

and the new element of monotony is 89.

that is

233. now question

sir uh every time this has a good magnum

fibonacci sequence

there's another way no matter formula

saying

using the your powerful calculator kai

and kain

sagutan without this table no it be

beginner any calculator

okay fibonacci number

general formula or we call this as

binet's formula

okay so anybody when it's formula so

this is

a formula that incorporates the golden

ratio so letting f sub

n is represent the n term

number of the fibonacci sequence where f

sub n

is equal to 1 over square root of 5

times okay the quantity the quantity of

1 plus square root of 5 over 2

raised to n minus one minus square root

of five over two raised to n

so using your calculator halimbawa

kanina

kinokua 13 annoying

canada hundred

thirty-three okay tingling at ten using

this formula

okay for example no so properly turning

nothing

and nothing on 13 so the answer will be

233

so using your calculator so using your

calculator

okay so papa

okay using your calculator

okay says

production bar so one then down arrow

so my calculator is about high square

root of five

and then okay

one parentheses we have then

again in fraction bar we have one plus

square root of five then down arrow we

have two

right side then enclose by parentheses

this one and then pin the thing

and then right arrow then minus

again open parenthesis and then your

production var

then one minus square root of five

then baba that is two

then close on it again and close

parenthesis

and then pin it in and that is thirteen

so you know and then again

parentheses okay click on

and that is 233.

let's say first 13 terms so using this

formula

so young f25 so papadi

uh so papadi

and 25

and then

by 10.25 so that is equals to

75 025 so that is equal to 7525 so you

can use this formula

next properties of fibonacci numbers so

first determine whether the statement 3

times f sub n minus f sub n minus 2

is equal to f sub n plus two for n is

greater than or equal to three so that

kappa f no no if your n is greater than

or equal to three

sudhapat capacitor nothing dito any

number

uh equal to greater than or equal to

three

to you so let's try

okay for example

okay uh start the k3

if your n is three so that will become

three plus two

so pivot and then substitute that in

three times

f sub three minus three minus two

penalty so that is f sub five

three times saying f sub three nothing

in value need to i two okay

f sub three two minus f sub one

so we all know that f sub five that is

five is equal to three times two this

is six minus one bucket one you know f

sub one attention

okay one so therefore five is equal to

five

and that is true therefore the statement

3 times f sub n minus f

sub n minus 2 is equal to f sub n plus 2

for n is greater than or equal to 3 is

true another

if n is even then f sub n is odd number

so check nothin

and an event number so i'll bring even

number two

four six eight ten and so on so

even numbered outing even number more

then ny even number then

f

so therefore that is wrong and

this statement if n is event and f sub n

is add number

this property is false

next okay thank you for watching this

video i

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