# data[].attributes.body
Ms. GRAZEL LANTINAY BALUAN❤

# data[].attributes.body
LET US ALL STAND AND FEEL THE PRESENCE OF THE LORD!

# data[].attributes.attachment_link.title
Active Maths - Directions Dance for kids

# data[].attributes.body
WHAT IS OUR TOPIC LAST TIME?

# data[].attributes.body
OUR TOPIC LAST TIME IS ALL ABOUT ARITHMETIC SEQUENCE.

It is important because the next lesson will delve into the application of arithmetic sequences, ensuring a solid foundation for the upcoming activities.

# data[].attributes.body
At the end of the lesson, the learners can:

a.     identify the common difference and nth term of an arithmetic sequence,

b.  apply their knowledge to find the nth term of arithmetic sequence using its formula; and

c. calculate the nth term of an arithmetic sequence with the help of Padlet.

# data[].attributes.body
TO HELP STUDENTS UNDERSTAND MORE ABOUT ARITHMETIC SEQUENCE!

# data[].attributes.headline
REAL LIFE SCENARIO OF ARITHMETIC SEQUENCE

# data[].attributes.subject
REAL LIFE SCENARIO OF ARITHMETIC SEQUENCE

# data[].attributes.body
- Questions:

- How can you relate the pattern in the Padlet arithmetic sequences to a real-life scenario?

- Why do you think understanding arithmetic sequences is valuable in solving real-world problems?

- How might businesses or industries use arithmetic sequences in their operations?

# data[].attributes.attachment_caption
THESE ARE THE STEPS IN SOLVING ARITHMETIC SEQUENCE.

# data[].attributes.headline
HOW TO FIND NTH TERM AND THE COMMON DIFFERENCE IN ARITHMETIC SEQUENCE?

# data[].attributes.subject
HOW TO FIND NTH TERM AND THE COMMON DIFFERENCE IN ARITHMETIC SEQUENCE?

# data[].attributes.body
The teacher will guide the students in analyzing the patterns and connect them to real-life scenarios. Discuss the significance of identifying these patterns in various contexts, fostering a deeper understanding of the practical applications of arithmetic sequences.

# data[].attributes.attachment_link.readable_text
Whether we're aware of it or not, one of the earliest concepts we learn in math fall under arithmetic sequences. When we count and observe numbers and even skip by $2$'s or $3$'s, we're actually reciting the most common arithmetic sequences that we know in our entire lives.
Arithmetic sequences are sequences of numbers that progress based on the common difference shared between two consecutive numbers.
This article will show you how to identify arithmetic sequences, predict the next terms of an arithmetic sequence, and construct formulas reflecting the arithmetic sequence shown. We'll also learn how to find the sum of a given arithmetic sequence.

Let's go ahead first and understand what makes up an arithmetic sequence.

What is an arithmetic sequence?

Arithmetic sequences are sequences of number that progress from one term to another by adding or subtracting a constant value (or also known as the common difference). Let's observe the two sequences shown below:

\begin{aligned} 2^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 4^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 7^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 10^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 13^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 16...\\ 34^{\color{green}\displaystyle{\overset{-2}{\rightarrow}}} 32^{\color{green}\displaystyle{\overset{-2}{\rightarrow}}} 30^{\color{green}\displaystyle{\overset{-2}{\rightarrow}}} 28^{\color{green}\displaystyle{\overset{-2}{\rightarrow}}} 26^{\color{green}\displaystyle{\overset{-2}{\rightarrow}}} 24...\end{aligned}

The two sequences are great examples of arithmetic sequences.

We can find the next term for the first sequence by adding $3$ to the previous term.

For the second one, we can see that the terms gradually decrease by $2$.

The next term can be determined for each sequence by adding (or subtracting) a constant, and the two are arithmetic sequences.

This means that any sequence that reflects a similar pattern will be considered an arithmetic sequence. Why don't we generalize this rule for all sequences, $\{a_1, a_2, ..., a_n\}$?

Arithmetic sequence definition

This means that when we have a sequence, $\{a_1, a_2, ..., a_n\}$, in general, it is an arithmetic sequence when:

\begin{aligned} a_2 - a_1 &= d\\ a_3 - a_2 &=d\\ .\\.\\.\\a_n - a_{n -1}& = d\end{aligned}

This means that a sequence is said to be an arithmetic sequence when the difference between two consecutive terms such as $a_1$ and $a_2$, $a_3$ and $a_4$ as well as $a_{n-1}$ and $a_n$ must all be the equal to a constant, $d$.

Let's go ahead and observe $5, 8, 11, 14, 17, 20, 23,...$.

We can observe the differences between each pair of consecutive terms to confirm if it is indeed an arithmetic sequence.

We can see that each consecutive term skip by $3$:

\begin{aligned} 5^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 8^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 11^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 14^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 17^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 20^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 23...\end{aligned}

This confirms the fact that the sequence is arithmetic.

How to solve arithmetic sequences?

We can find the next term of an arithmetic sequence and even determine the specific formula for a given sequence using the arithmetic sequence's recursive and explicit rules.

The two forms have their own strengths, and knowing when to use the two can help a lot in addressing what's needed in a given problem involving arithmetic sequences.

Arithmetic sequence formula

Let's say we $a_1$ represents the first term of the sequence, $a_n$ be the last term of the sequence, and $d$ be the common difference shared between two consecutive terms from the terms; we have the following arithmetic sequence formulas.
Recursive Rule
\begin{aligned}a_n = a_{n-1} + d\end{aligned}

This form of the arithmetic sequence formula is most helpful when we're looking for the $n$th term of a given arithmetic sequence.

For example, if we have $-5, 0, 5, 10, 15, 20...$, we can use the recursive formula to find the next term.

The common difference shared by two consecutive terms will always be $5$.

This means that the seventh term, $a_7$, will be equal to $20 + 5 = 25$.

Explicit Rule
\begin{aligned}a_n = a_1 +(n -1) d\end{aligned}

This form is most helpful if we want to find the formula representing the arithmetic sequence. Using the same example, $-5, 0, 5, 10, 15, 20...$, we can use the explicit rule to find the sequence's general rule.

We have $a_1 = -5$ and $d = 5$, so we can replace the general rule and simplify the right-hand side to find the general rule for $n$.

\begin{aligned} a_n &= -5 + (n- 1)5\\&= -5 + 5n - 5\\&=5n - 10\end{aligned}

With the explicit rule or form, we can also predict the $100$th value of the sequence by substituting $ n = 100$ into the formula.

\begin{aligned} a_{100} &=5(100) - 10\\&= 500 - 10\\&= 490\end{aligned}

How to find the sum of the arithmetic sequence?

When dealing with arithmetic sequences, it's also helpful to learn about arithmetic series.

The only difference between arithmetic sequences and series is that arithmetic series reflects the sum of an arithmetic sequence.

We can find the sum of an arithmetic sequence or the value of an arithmetic series by finding the average of the first and the last term then multiplying the result by the number of terms.

This means that given $a_1$ and $a_n$, the sum of the sequence (or the value of the arithmetic series) is equal to $S_n = \dfrac{n(a_1 + a_n)}{2}$.

Let's say we want to find the sum of the series, $7 + 14 + 21 + ... +70 + 77$, it's important that we check if the series is indeed an arithmetic series

\begin{aligned} 7^{\color{green}\displaystyle{\overset{+7}{\rightarrow}}} 14^{\color{green}\displaystyle{\overset{+7}{\rightarrow}}} 21^{\color{green}\displaystyle{\overset{+7}{\rightarrow}}} ... 63^{\color{green}\displaystyle{\overset{+7}{\rightarrow}}} 77\end{aligned}

We have confirmed that the sequence is arithmetic and $d = 7$, so we can use the formula for the sum of an arithmetic series by first finding the value of $n$.

Using the explicit rule of an arithmetic sequence, we have the following:

\begin{aligned} a_n &= a_1 + (n-1)d\\77&= 7 + (n-1)7\\11&=1 + (n-1)\\n&=11\end{aligned}

Now that we have $a_1 = 7$, $a_n = 77$, and $n= 11$, we can use the sum formula to find the value of the arithmetic series.

\begin{aligned} S_n &= \dfrac{n(a_1 + a_n)}{2}\\&= \dfrac{11(7 + 77)}{2}\\&= \dfrac{11(84)}{2}\\&= 11(42)\\&= 462\end{aligned}

This means that we can use the different elements and formulas involving the arithmetic sequence to solve different types of problems. Make sure to review the different rules and formulas before trying out these problems shown below!
Example 1
Determine whether the sequences below are arithmetic sequences or not.

a. $3, 9, 15, ..., 75, 81$
b. $-12, -6, 0, ..., 60, 66$
c. $2, 4, 8, ..., 256, 512$
d. $2, 5, 8, ..., 71, 75$

To confirm whether a sequence is an arithmetic sequence, we can check the sequences' patterns and see if each pair of consecutive terms shares a common difference. If they do, the sequence is considered an arithmetic sequence.

\begin{aligned} 3^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} 9^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} 15^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} ... 75^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} 81\end{aligned}

We can see that the first sequence share a common difference of $6$, so the given sequence is an arithmetic sequence.

Applying the same process for the next sequence, we have the following:

\begin{aligned} -12^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} -6^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} 0^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} ... 60^{\color{green}\displaystyle{\overset{+6}{\rightarrow}}} 66\end{aligned}

Hence, the second sequence is also an arithmetic sequence.

\begin{aligned} 2^{\color{green}\displaystyle{\overset{+2}{\rightarrow}}} -4^{\color{green}\displaystyle{\overset{+4}{\rightarrow}}} 8^{\color{green}\displaystyle{\overset{+8}{\rightarrow}}} ... 256^{\color{green}\displaystyle{\overset{+256}{\rightarrow}}} 512\end{aligned}

The differences between the pairs of consecutive terms vary, so the sequence is not an arithmetic sequence.

Let's now take a look at the last sequence:

\begin{aligned} 2^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} -5^{\color{green}\displaystyle{\overset{+3}{\rightarrow}}} 8^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} ... 71^{\color{green}\displaystyle{\overset{+4}{\rightarrow}}} 75\end{aligned}

Since the pairs of consecutive terms have varying differences, so the sequence is not an arithmetic sequence.
Example 2
What is the common difference of the arithmetic sequence, $-9, -3, 3, 9, ...$? Find the seventh term of the sequence.

We can start by observing the common difference shared by each pair of consecutive terms.

This means that the arithmetic sequence has a common difference of $6$. We can use the common difference to find the next terms by adding each term by $6$ until we have the seventh term.

\begin{aligned} a_4&= 9\\ a_5&= 9+ 6\\&= 15\\a_6 &= 15 + 6\\&= 21\\a_7 &= 21 + 6\\&= 27\end{aligned}

This means that the seventh term of the arithmetic sequence is $27$.
Example 3
Find an equation that represents the general term, $a_n$, of the given arithmetic sequence, $12, 6, 0, -6, -12, ...$. Use this equation to find the $100$th term of the sequence.

We can begin by finding the common difference of the arithmetic sequence.

\begin{aligned} 12^{\color{green}\displaystyle{\overset{-6}{\rightarrow}}} 6^{\color{green}\displaystyle{\overset{-6}{\rightarrow}}} 0^{\color{green}\displaystyle{\overset{-6}{\rightarrow}}} -6^{\color{green}\displaystyle{\overset{-6}{\rightarrow}}} -12\end{aligned}

From this, we can see that the sequence has a common difference, $d$, of $-6$. To find the equation representing the $n$th term of the sequence, we can use the explicit rule for the arithmetic sequence, as shown below.

\begin{aligned} a_n &= a_1 + (n-1)d\\&= 12 + (n - 1)(-6)\\&= 12 + (-6n + 6)\\&= 12 - 6n + 6\\&= 18 - 6n\end{aligned}

Now that we have the equation for the $n$th term's general form, we substitute $n = 100$ into the equation to find the hundredth term.

\begin{aligned} a_{100}&= 18 - 6(100)\\&=18 - 600\\&= -582\end{aligned}

This means that the general form of sequence's $n$th term is $-6n + 18$, and the $100$th term of the sequence is $-582$.
Example 4
What is the sum of the series $6 + 11 + 16 + 21 + ... +216+ 221$?

\begin{aligned} 6^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} 11^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} 16^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} 21^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} ...,216^{\color{green}\displaystyle{\overset{+5}{\rightarrow}}} 221\end{aligned}

Observing the sequence, we can see that the common difference, $d$, of the series is equal to $5$.

To find the sum of the arithmetic series, we'll need the first and last terms and the number of terms within the series. Let's use the explicit rule, $a_n = a_1 + (n - 1)d$, to find the value of $n$ first.

\begin{aligned} a_1 &= 6\\a_n&= 221\\d&= 5\\\\221 &= 6 + (n-1)5\\215&=5(n - 1)\\43&= n - 1\\ n&= 44\end{aligned}

This means that the series has $44$ terms. We can now use the formula for the sum of an arithmetic series since we have $a_1$, $a_n$, and $n$ already.

\begin{aligned} S_n &= \dfrac{n(a_1 + a_n)}{2}\\&=\dfrac{44(6 + 221)}{2}\\&= \dfrac{44(227)}{2}\\&= 22(227)\\&= 4994\end{aligned}

Hence, the arithmetic series has a sum of $4994$.
Example 5
An outdoor amphitheater contains $12$ seats on its first row, $14$ seats on the second row, $16$ seats on the third row, etc. If the amphitheater has twelve rows, how can seats can it accommodate?

As we can observe, the outdoor amphitheater increases gradually over each row.

\begin{aligned} 12^{\color{green}\displaystyle{\overset{+2}{\rightarrow}}} 14^{\color{green}\displaystyle{\overset{+2}{\rightarrow}}} 16...\end{aligned}

Since each pair of consecutive terms has $2$ as its common difference, we can say that the number of seats for all eighteen rows will reflect an arithmetic sequence.

We'll need to know the last term of the sequence first, and we can use the explicit rule to find the number of seats in the $18$th row.

\begin{aligned} a_n &= a_1 + (n-1)d \\ a_{18} &= 12 + (18 - 1)2\\&= 12 + 34\\&= 46 \end{aligned}

This means that the $18$th row has $46$ seats. To find the total number of seats, we can find the sum of the entire sequence (or the arithmetic series) using the formula, $S_n = \dfrac{n(a_1 + a_n)}{2}$.

\begin{aligned} 12 + 14 + 16 + ... + 46 &= S_n\\&=\dfrac{18(12 + 46)}{2}\\&= \dfrac{18(58)}{2}\\&=9 (58)\\&= 522\end{aligned}

This means that the outdoor amphitheater has a total seat capacity of $522$.

# data[].attributes.attachment_link.title
Arithmetic sequence - Pattern, Formula, and Explanation

# data[].attributes.attachment_link.description
Arithmetic sequences are sequences that progress by adding the same constant each time. Know more about their behavior and properties here!

# data[].attributes.body
IMPORTANCE OF ARITHMETIC SEQUENCE!

# data[].attributes.attachment_link.readable_text
What is the importance of arithmetic sequence in our daily life?

The arithmetic sequence is important in real life because this enables us to understand things with the use of patterns. An arithmetic sequence is a great foundation in describing several things like time which has a common difference of 1 hour. An arithmetic sequence is also important in simulating systematic events.
What is arithmetic progression used for?
What is the use of Arithmetic Progression? An arithmetic progression is a series which has consecutive terms having a common difference between the terms as a constant value. It is used to generalise a set of patterns, that we observe in our day to day life.

What are the applications of arithmetic mean?

The arithmetic mean is used frequently not only in mathematics and statistics but also in fields such as economics, sociology, and history. For example, per capita income is the arithmetic mean income of a nation's population.
What are the types of arithmetic mean?
Mean is the most commonly used measure of central tendency. There are different types of mean, viz. arithmetic mean, weighted mean, geometric mean (GM) and harmonic mean (HM). If mentioned without an adjective (as mean), it generally refers to the arithmetic mean.

What are the 2 types of arithmetic mean?

It is referred to as the ratio of the total sum of given observations to the total number of observations. The arithmetic mean can be positive, negative, or zero. There are two types of Arithmetic Mean, Simple Arithmetic Mean. Weighted Arithmetic Mean.
What is difference between arithmetic mean and average?
The difference between Arithmetic mean and Average. When used as nouns, arithmetic mean means the measure of central tendency of a set of values computed by dividing the sum of the values by their number, whereas average means the arithmetic mean.

What is average formula?

The formula to calculate average of given numbers is equal to the sum of all the values divided by total number of values. Average = (Sum of Values/Number of Values)
Is Mean and median the same?
The mean is the average of a data set. The mode is the most common number in a data set. The median is the middle of the set of numbers.

Which is better mean and median?

Unlike the mean, the median value doesn't depend on all the values in the dataset. Consequently, when some of the values are more extreme, the effect on the median is smaller. When you have a skewed distribution, the median is a better measure of central tendency than the mean.
Why use mean instead of median?
When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. The more skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean.

How do you compare mean and median?

A mean is computed by adding up all the values and dividing that score by the number of values. The Median is the number found at the exact middle of the set of values. A median can be computed by listing all numbers in ascending order and then locating the number in the centre of that distribution.
How do different distributions affect mean and median?
To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.

What is the relationship between mean and median?

Mean is the average of all the values. Median is the middle value, dividing the number of data into 2 halves. In other words, 50% of the observations is below the median and 50% of the observations are above the median. Mode is the most common value among the given observations.
How do you compare the mean and median in a histogram?
Here are some tips for connecting the shape of a histogram with the mean and median:

If the histogram is skewed right, the mean is greater than the median.

If the histogram is close to symmetric, then the mean and median are close to each other.

If the histogram is skewed left, the mean is less than the median.

How do you compare distributions?

The simplest way to compare two distributions is via the Z-test. The error in the mean is calculated by dividing the dispersion by the square root of the number of data points. In the above diagram, there is some population mean that is the true intrinsic mean value for that population.
Are mean median and mode equal in normal distribution?
The normal distribution is a symmetrical, bell-shaped distribution in which the mean, median and mode are all equal. It is a central component of inferential statistics. The standard normal distribution is a normal distribution represented in z scores.

Can a data set have the same mean median and mode?

A data set can have the same mean, median, and mode. For a symmetric distribution, the standard deviation and mean are equal. false- MEDIAN and mean are equal.
Does the mode represent the center of the data?
The mode(s) does (do) not represent the center because it is the smallest data value.

Does the median represent the center of the data?

The median is the value in the center of the data. Half of the values are less than the median and half of the values are more than the median. It is probably the best measure of center to use in a skewed distribution.
Will a data set always have exactly one mode?
A data set will always have exactly one mode. The 75th percentile is the value that separates the lower 75% of data from the upper 25%.

Are the characteristics of the individuals of the population being studied?

Variables are the characteristics of the individuals of the population being studied.
Why the median resistant but the mean is not?
Why is the median​ resistant, but the mean is​ not? The mean is not resistant because when data are​ skewed, there are extreme values in the​ tail, which tend to pull the mean in the direction of the tail.

Which measure of central tendency best describes the weight of the candy?

Which measure of central tendency best describes the weight of the candy? Adjust the window O A. Ymax=4 Is the histogram for the data set skewed right, skewed left, or symmetric? The median weight of the candies is grams.
What is the mode of a list?
The average is one number in a set of numbers that is somehow typical of the whole set of numbers. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

What is the preferred measure of central tendency?

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. For data from skewed distributions, the median is better than the mean because it isn't influenced by extremely large values.
Which center of Measure works better here?
The median works better here since it's more typical of most of the data.

Which average is best to use?

Mean is the most frequently used measure of central tendency and generally considered the best measure of it. However, there are some situations where either median or mode are preferred. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data.

# data[].attributes.attachment_link.title
What is the importance of arithmetic sequence in our daily life?

# data[].attributes.attachment_link.description
What is the importance of arithmetic sequence in our daily life? The arithmetic sequence is important in real life because this enables us to understand things with the use of patterns. An arithmetic sequence is a great foundation in describing several things like time which has a common difference of 1 hour.

# data[].attributes.body
THE STUDENTS WILL ANSWER THE FOLLOWING PROBLEMS IN ORDER FOR THEM TO UNDERSTAND MORE ABOUT ARITHMETIC SEQUENCE.

# data[].attributes.body
SOLVE AND APPLY WHAT YOU HAVE LEARNED!

# data[].attributes.wish_content.attachment_props.question
Which of the following is the Arithmetic sequence? 

# data[].attributes.wish_content.attachment_props.question
What is the next term of 3, 9, 15, 21, _ ?

# data[].attributes.wish_content.attachment_props.question
(3, 5, 7, 9, 11) Is this an example of arithmetic sequence? 

# data[].attributes.body
This video explains how arithmetic sequences are used in a specific real-life scenario that is not covered in class.

# data[].attributes.attachment_link.title
Real-World Problem about arithmetic sequence

# data[].attributes.wish_content.attachment_props.question
Are there any questions or Clarifications about our topic for today? 