Steps to solve
Start with drawing a table or any other recognition technique to help you understand what the sequence is adding or subtracting by
Understand and uncover what the sequence is changing and check to see if it matches the other numbers.
Next, create a fundamental equation for the sequence using tn = n and adding the variable of which the code is increasing; by for example, if the sequence was 2,4,6,8 since the series is going up by two each time, a possible equation could be t(n)=2n
Finally, a check could be helpful to see if the equation you created works and is correct. The check code is done by taking your equation, adding one of the starting terms from the given sequences into your code, and solving it to see if the answer is correct.
EXAMPLE PROBLEM ONE: Find the formula for the nth term
8,9,10,12,14
Step 1:
n t(n)
1 8
2 10
3 12
4 14
Step 2(could be mentally but should be thought of):
The sequence goes up by two each time.
Step 3:
Base Formula
t(n) = n
Changed for function:
t(n) = 2n+6
Step 4:
Check:
We can check our formula using one of the already given values from the table to see if we get the correct one. t(n)
Start with writing the formula you created and implementing a value from the table
t(2) = 2(2)+6
t(2) = 4+6
t(2) = 10
= The formula checks out
EXAMPLE PROBLEM TWO: Given the sequence, find the 1st, 3rd and 7th term using the formula tn = 5n + 7
Here we are given a formula and are told to find the first, third and seven.
Start with replacing the n value with the # and solve the rest.
a) tn = 5(1) + 7 b) tn = 5(3)+7 c) tn = 5(7)+7
tn = 5+7 tn = 15+7 tn = 35+7
tn = 12 tn = 22 tn = 42
EXAMPLE PROBLEM THREE
To help us understand the actual usefulness of explicit formulas, we can do some higher calculations with the formulas. Find t55 and t89 using the following sequence of 8,16,24,32,40
n tn
1 8
2 16
3 24
4 32
5 40
Formula is:
tn = 8n
Check:
tn = 8(4)
tn = 32
a)
t(55):
t(55) = 8(55)
t(55) = 440
b)
t(89) = 8(89)
t(89) = 712
General formula which helps find the nth term:
tn = a + (n-1)d
The recursion formula is a formula which defines any term in a sequence using previous terms
tn = tn-1 + d
Steps
Step 1
Locate the variables given to you in the question
Step 2
Input the found variables into the correct formula
Step 3
Solve for the necessary missing variable
EXAMPLE PROBLEM 1
Find t20 using the sequence 90,80,70,60,50,40
Step 1
Find the given variables
a = 90
d = -10
n = 20
t(20) = ?
Step 2
Since we are solving for an unknown term we can use the general term
tn = a + (n-1)d
t(20) = 90 + (20-1)(-10)
Step 3
Solve for the missing variable
t(20) = 90 + (19)(-10)
t(20) = 90 + (-190)
t(20) = -100
Formulas:
sn = n[2a + (n-1)d] / 2
And
sn = a * (rn-1)/r-1
*Also look at the yellow and dotted green lines *
Steps
Step 1
Locate the variables given to you in the question
Step 2
Input the found variables into the correct formula
Step 3
Solve for the necessary missing variable
EXAMPLE PROBLEM
Find the sum of this geometric series using this sequence 1/49 + 1/7 + 1 + 7+..............+40353607.
Step 1
Sn =?
n = ?
a = 1/49
r = 7
Step 2
Since we have two variables missing we need to use the general formula to solve for the missing n variable
Tn = a * rn-1
40353607 = (1/49)7N-1
Step 3
40353607 / 1/49 = 7n-1
1977326743 = 7n-1
711 = 7 n-1
11 = n-1
10 = n
Now that you only have a single missing variable you can solve for it.
Sn = 1/49 (710-1)/ 7-1
Sn = 960800.1633
Steps:
Step 1
Look for the t value or any other values given to you in the problem. Some might be obvious such as t1 = 2, while others might be harder to find.
Step 2
Once the given value is found, there will also usually be either a recursion formula or a constant the table changes which is what you need to find next.
Step 3
Once you have the formula, either by making it or giving it to you, you need to answer the question by implementing the values into the formula.
EXAMPLE PROBLEM ONE: Solve for the first three terms
t1 = 6
Formula:
tn = tn-1 -12
Step 1
In these questions, we are given the first t value of t1 = 6
Step 2
It is also known that we are given the formula for the question, which is seen to be tn = tn-1 -12
Step 3
Solve the necessary question.
t(2) = t2-1-12
t(2) = t1 -12
*here, we can see that t1 reapers into the equation, and knowing what t1 is since it was given ot us, we can input that into the equation.*
t(2) = 6-12
t(2) = -6
*Term 2 is -6*
Repeat step 3 for the rest of the terms
t(3) = t3-1-12
t(3) = t2 -12
t(3) = -6-12
t(3) = -18
t(4) = t4-1-12
t(4) = t3-12
t(4) = -18-12
t(4) = -30
EXAMPLE PROBLEM TWO
There are 15 rows in a movie theatre. The first row has twelve, the second row has 24, and the third has 36. How many seats does the third last row have?
Step1
Here seeing that the first three rows are given we are able to gather all of those t values
t1=12
t2=24
t3=36
Step2
Finding the formula
t(n) =(nx12)+12
We known from the first three terms that the sequence is going up by 12 and that the adding 12 to the previous value will give us the next hence adding 12 to 12 time to the number
Step 3
Solve
Now that we have the t value and the formula we can solve for what the question is asking for
t(12) = (12x12)+12
t(12) = 144 + 12
t(12) = 156
The third last row has 156 seats
EXAMPLE PROBLEM THREE solve for the first 2 terms
t1 = 56
Formula:
tn = 82(tn-1)
t(2)= 82(t2-1)
t(2) = 82(56)
t(2) = 4,592
t(3) = 82( t3-1)
t(3) = 82(t2)
t(3) = 82(4,592)
t(3) = 376,544
All terms can be represented by tn,r and this coils be represented by the formula
tn,r = tn-1,r-1 + tn-1,r
Steps
Step 1
Start off by drawing the pascal triangle. You can go up to the row needed or add some extra just in case.
Step 2
Based on what the question is asking locate the rows and column you will be needing in this problem
Step 3
Solve the question
EXAMPLE PROBLEM 1 Find the value of these coordinates using the Pascal Triangle.
a) t3,3
Step 1
The question is asking for the 3rd row
And the 3rd column so start with making a
Pascal triangle with these requirements
1
1 + 1
1 + 2 + 1
1 + 3 + 3 + 1
Step 2
using the created Triangle we can locate
Row 3 and column 3 in that row
Step 3
Now we can solve
t3,4 = 1
b) t6,2
Step 1
The question is asking for the 6th row
And the 2nd column so start with making a Pascal triangle with these requirements
Step 2
using the created Triangle we can locate using the now made triangle, we can
Row 6 and column 2 and locate the row and column in the question.
1
1 + 1
1 + 2 + 1
1 + 3 + 3 + 1
1 + 4 + 6 + 4 + 1
1 + 5 + 10 + 10 + 5 + 1
1 + 6 + 15 + 20 + 15 + 6 + 1
Step 3
Now we can solve t6,2 = 15
EXAMPLE PROBLEM 2 Expand and simplify
(x + y)6
Step 1
1
1 + 1
1 + 2 + 1
1 + 3 + 3 + 1
1 + 4 + 6 + 4 + 1
1 + 5 + 10 + 10 + 5 + 1
1 + 6 + 15 + 20 + 15 + 6 + 1
Step 2
Now we can see that to expand we will be needing row 6’s coefficients
Step 3
(x + y) 6
= 1 x6y0 + 6 x5y1 + 15 x4y2 + 20 x3y3 + 15 x2y4 + 6 x1y5 + 1 x0y6
In geometric sequences a few formulas are needed to solve some problems.
General formula: helps find the nth term from a series of numbers
tn = a * rn-1
Recursion formula:
tn = r * tn-1
Steps
Step 1
Locate the variables given to you in the question
Step 2
Input the found variables into the correct formula
Step 3
Solve for the necessary missing variable
EXAMPLE PROBLEM
Find t40 with the sequence 7776, 1296, 216, 36, 6
Step 1
a = 7776
r = divided by 6 but since only multiplication x1/6
Step 2
We will use the general formula as we are finding the nth term
tn = a * rn-1
t(40) = 7776 * 1/640-1
Step 3
Solve
t(40) = 1/1296
➕➖✖️➗ Sequences and Series
Hello Guest!
This mind map contains all the necessary steps to help you create your own podcast.
- RIGHT SIDE: you will create the concept.
- LEFT SIDE: you will set up all the practical elements.
Important elements
Now that you have a clear idea about the concept of your new podcast, it's time to focus on some practical elements.
Welcome to lesson 6, our final lesson on the Series and Sequences unit. In our final unit we will be learning different ways to solve Arithmetic and Geometric series questions
Variables need Geometric follow the solid lines
Variables needed Arithmetic Follow The dotted lines
Add your cover image
Upload it here
Your podcast needs cover art.
This is a cover image that should represent your podcast and also stand out.
Follow the guidelines:
- 1:1 aspect ratio (square format)
- 3000 x 3000 pixels.
Geometric Formula
This is the structure of the episode.
It's important to have an outline or some steps that will apply to each individual episode.
Build your own format by choosing from these elements:
TeaserIntro musicWelcomeShort summaryInterviewThe ideas presentedQ&AAd spotCall to action (Follow/like)Outro music
Welcome to lesson 5 of the sequences and series unit. In this lesson we will be going over Geometric sequences. As always lets start this lesson with understanding what we are learning.
Geometric sequences:
A geometric sequence is a series of numbers in which each term is found by multiplying the previous term by a constant factor
r:
The variables “r” is the constant factor in which the term is bing multiplied by each time.
a:
The variable “a” is the first term in a series of numbers
Variables used
tn = is the term number and n is the certain number in a series of numbers
Arithmetic Sequences
You are your own 'stage director'.
Think about how you want to structure your show.
Welcome to lesson 4 of the sequences and series unit. In This lesson, we will be going over the Arithmetic Sequences.
Let's start with understanding what Arithmetic sequences actually are and the lesson and formulas needed in this lesson
Arithmetic Sequences:
Arithmetic sequences are a series of numbers which are increasing or decreasing by the same number each time.
Many experts are showing different researches about the optimum time, and you can take the advice, for example:
- 28 minutes podcasts - because that is the average driving time
- less than one hour - because that is the maximum time a person can pay attention
However, you can also set a time you are comfortable with.
d:
The variable “d” is what the series of numbers is increasing or decreasing by.
Example 2,4,6,8,10
a = 2
d= 2 as the series is increasing by two each time so the “d” value is positive 2
a:
The first term in a sequence is represented by the variable “a”
For example 1,2,3,4,5,6,7,8
In this series of numbers, the “a” would be 1 as it's the first term.
n:
The number in a series
Variables used:
tn:
term number of the sequence
Pascal's Triangle
These 3 questions will help you create a successful podcast.
Think about each of them and write the answer.
Welcome to lesson three of the series and sequences unit. In this lesson, we will be discussing Pascal’s Triangle. Let’s start by exploring and understanding the terms and definitions we will use in today’s lesson.
Let's start with understanding what the pascal triangle is.
The Pascal Triangle:
To create a triangle consisting of a border of ones and adding the inside of each row
Example of a Pascal Triangle:
1
1 + 1
1 + 2 + 1
1 + 3 + 3 + 1
1 + 4 + 6 + 4 + 1
And so on…………
Find your unique asset
What can you do differently than other people who have a podcast?
t:
The t represents the term
Example:
t2,3 =
r:
The variable r represents the column number and each triangle starts at column 0
The variable we are going to be using in todats lesson are
n:
The variable n represents the row number and each triangle starts at row 0
Recursion Formula
After you added the topic, think about a good name for your podcast.
Choose a unique name that has related words to the topic.
At the same time, focus on more broad words to keep the possibility to expand in the future.
Try to keep it short to be easily remembered by people.
Welcome to lesson two of our series and sequences unit. In this lesson, we will be going over the recursion formula and some examples of the recursion formula.
Let's start with understanding the recursion formula and the variables we will use for this unit.
Recursion Formula:
A recursive formula uses the last or set of previous terms to define each term in a sequence.
Variables:
Yes, any variables that you find easy to use or comfortable with you are free to use but for this lesson, I will be using the variables t and n
Just like the last lesson, these variables mean:
n: A lowercase n variable is used to label the number in a sequence
Explicit Formula
The first thing you should start with is choosing a topic or a niche.
Try to think about something you can talk about for many episodes.
What do you want to talk about in your podcast?
What is the main subject you are going to share with your audience?
Welcome to lesson 1 of this new unit, Sequences and Series. In this new unit, we will go over explicit formulas, understand how to solve them, and understand what they mean.
Let’s start with what sequences and explicit formulas are.
Sequences
Definition: A series of numbers in a list separated by a to-be-determined repeating amount
Example: 1,2,3,4,5,6,7,8,9 Here, the list is separated by +1
Explicit Formulas
Definition: An explicit formula calculates the unknown term in a sequence
Term
Definition: an individual number in the sequence
Example: 1,2,3,4,5,6,7,8,9 Here, a possible term is 5
t: A lowercase t variable is used to label the term
Variables used
n: A lowercase n variable is used to label the number in a sequence
